Theodosius <Bithynius>; Clavius, Christoph
,
Theodosii Tripolitae Sphaericorum libri tres
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arcus CD. </
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<
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xml:space
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">Item vt AF, ſinus rectus arcus BD, ad AD, ſinum totum, ita AC,
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ſinus totus ad AE, ſecantem arcus CD, qui complementum eſt eiuſdem arcus
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BD. </
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<
s
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xml:space
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">Quare ſinus totus medius proportionalis eſt inter AF, ſinũ cõ plementi
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arcus CD, & </
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<
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xml:space
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Item inter AF, ſinum rectum arcus BD, & </
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<
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xml:space
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ſecantem complementi eiuſdem arcus BD. </
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<
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eſt tertium.</
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<
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xml:space
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">POSTREMO, quia triangula ACE;
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</
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<
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">GBA, æquiangula ſunt, quòd anguli C, B, ſint
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recti; </
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<
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">& </
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<
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xml:space
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">alterni CAE, BGA, nec non alterni
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">29. primi.
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4. ſexti.</
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AEC, GAB, æquales: </
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<
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xml:space
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">erit vt CE, tangens ar-
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cus CD, ad AC, ſinum totum, ita AB, ſinus to-
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tus ad BG, tangentem complementi eiuſdem ar-
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cus CD. </
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<
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xml:space
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">Pari ratione erit, vt BG, tangens arcus
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BD, ad AB, ſinum totum, ita AC, ſinus totus
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ad CE, tangentem complementi eiuſdem arcus BD. </
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<
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</
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<
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xml:space
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">Quã ergo proportionem habet ſinus complementi arcus cuiuſuis, &</
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<
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erat demonſtrandum.</
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">ITAQVE per ea, quæ primo loco demonſtrata ſunt in hac propoſ. </
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<
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xml:space
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">ſi fiat, vt ſin{us} com
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xlink:label
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xlink:href
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xml:space
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">Quo pacto
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tangentes
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omniú ar-
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cuum repe-
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tiantur.</
note
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plementi cuiuſuis arc{us} ad ſinũ rectũ eiuſdẽ arc{us}, ita ſin{us} tot{us} ad aliud; </
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<
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xml:id
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xml:space
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">hoc eſt, ſi ſi-
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n{us} arc{us} cui{us}libet in ſinum totũ multiplicetur, (quod fiet facile, ſi ei ad dextrã præ-
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ponas tot ci fras, quot in ſinu toto continentur, nempe ſeptem, ſi ſin{us} tot{us} fuerit
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10000000. </
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<
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xml:space
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">vel quinque, ſi ſinus totus fuerit 100000.) </
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<
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">productusq́; </
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<
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">numerus per ſi-
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num complementi eiuſdem arc{us} diuidatur: </
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<
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">inuenietur Tangens illi{us} arc{us}, cui{us} ſi-
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num complementi accepiſti, vel cuius ſinum rectum in ſinum totum multiplicaſti. </
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<
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ſi tangens arc{us} gra. </
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<
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<
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">quæratur, adiungemus eius ſinui recto 5000000 ſeptem ci-
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fras, hoc modo. </
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<
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">50000000000000. </
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<
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">& </
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<
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xml:space
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">hunc numerum per 8660254 ſinumcomple-
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menti eiuſdem arcus grad. </
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<
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<
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">partiemur. </
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<
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xml:space
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">Nam quotiens numerus 5773503. </
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tangentem arcus grad. </
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<
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<
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">quatenus ſinus totus eſt 10000000. </
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<
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">Hinc fit, tangentes
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<
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ſione om
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nes tangen
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tes eliciun-
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tur.</
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emnes per ſolam diuiſionem inueniri. </
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<
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">Nam ſiomnium arcuum ſinubus, initio facto à
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principio quadr antis, ſeptem ci fr{as} appon{as}, & </
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<
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">compoſitos numeros per ſinus comple
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mentorum eorundem arcuum, quemlibet per ſuum correſpondentem, diuid{as}, prodi-
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bunt omnium arcuum tangentes, vt ex demonſtratis liquido cõſtat. </
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<
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facilis eſt conſtructio tabulæ Tangentium.</
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<
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<
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<
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xlink:label
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">Qua tõne
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ſecátes om-
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niú arcuũ
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inueſtigen-
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tur.</
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nus complementi cuiuſuis arcus ad ſinum totum, ita ſinus totus ad aliud: </
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<
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xml:id
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xml:space
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nus totus in ſeipſum multiplicetur, (quod facile fiet, ſi ei ad dextram præpon{as} tot
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cifras, quot ſunt in ſinu toto, puta ſeptem, vel quinque, prout ſinus totus habuerit
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ſeptem, aut quinque cifras.) </
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<
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<
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cus diuidatur: </
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<
s
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cta eſt. </
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<
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<
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<
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<
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">(pro-
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ductum ſcilicet numerum ex ſinu toto in ſeipſum) per 8660254. </
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<
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arcus grad. </
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<
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<
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<
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</
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<
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<
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