Theodosius <Bithynius>; Clavius, Christoph
,
Theodosii Tripolitae Sphaericorum libri tres
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ſinus CH, cuius radix quadrata notum dabit ſinum rectum CH. </
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<
s
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xml:space
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">Eademq́ue ratio eſt
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de cæteris.</
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<
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<
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xml:space
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">IDEM hac etiam ratione oſtendi poteſt. </
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<
s
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echoid-s4230
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xml:space
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">Quoniam enim EF, ſinus rectus arcus
<
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CE, notus ponitur, cognoſcetur & </
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>
<
s
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">EI, ſinus complementi ei uſd em arcus, hoc eſt, re@
<
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note-125-01
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">3. huius.</
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cta AF, illi æqualis. </
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<
s
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echoid-s4232
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xml:space
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">Detracta igitur recta AF, hoc eſt, ſinu complementi arcus CE,
<
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<
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xlink:label
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note-125-02
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note-125-02a
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">34. primi.</
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>
ex ſinu toto AC, cognitus erit ſinus verſus FC, arcus eiuſdem CE, vt etiam in coroll.
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</
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<
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<
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<
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">oſtendinaus. </
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<
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">Quia vero quadratum rectæ CE, æquale eſt quadratis rectarum EF,
<
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FC; </
s
>
<
s
xml:id
="
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xml:space
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">fit, vt quadrata rectarum EF, FC, nota rum in vnam ſummam collecta efficiant
<
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quadratum rectæ CE: </
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>
<
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">cuius radix quadrata ipſam rectam CE, reddet notam; </
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<
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">ac proinde
<
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huius radicis dimidium dabit CH, ſinum rectum arcus CD, qui dimidium eſt dati arcus
<
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CE, notum.</
s
>
<
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<
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<
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">VICISSIM ex hac eadem propof. </
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">4. </
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<
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">colligitur, ſi ſinus rectus alicuius arcus cognitus
<
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">Ex ſinu re-
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cto cuiuſ-
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uis arcꝰ co-
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gnito notꝰ
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ſit ſinus re
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ctus alteriꝰ
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arcꝰ, qui il-
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liꝰ ſit duplꝰ
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dummodo
<
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quadrante
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minor ſit.</
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ſit, notum etiam fieri ſinum rectum alrerius arcus, qui illius duplus ſit, dummodo quadran-
<
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te ſit minor: </
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<
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">ita vt ex CH, ſinu recto arcus CD, cognito cognoſcatur etiam EF, ſinus rectus
<
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arcus CE, qui arcus CD, eſt duplus. </
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<
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">Cum enim ſin us CH, ſit medio loco proportionalis in-
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ter medictatem ſinus totius, & </
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<
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">ſinum verſum FC, vt oſtendimus; </
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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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xml:space
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">ſinu verſo FC, contentum æquale quadrato ſinus recti CH. </
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<
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">Quare
<
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quadratum ſinus CH, noti erit illud rectangulum; </
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<
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xml:space
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">quo diuiſo per dimidium ſinus totius,
<
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notus euadet ſinus verſus FC. </
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>
<
s
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xml:space
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">Quia vero recta CE, cum ſit dupla ſinus CH, noti nota eſt,
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erit & </
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<
s
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">eius quadratum notum: </
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<
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">à quo ſi auferatur quadratum ſinus verſi FC, noti, relinque-
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tur etiam quadratum rectæ EF, notum; </
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<
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xml:space
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">(cum quadratũ rectæ CE, quadratis rectarum CF,
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FE, ſit æquale.) </
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<
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">ac proinde radix quadrata illius notum dabit ſinum rectum EF.</
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, _ſemidiametrum_ AC, _ſecet bifariam, vt_
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_in hac figura contingit, erit adhuc_ CH, _ſinus arcus_ CD,
<
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<
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125-01
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xlink:href
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/125-01
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_medio loco proporlionalis inter_
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, _ſemiſſem ſinus totius,_
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& </
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<
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<
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, _ſinũ verſum arcus_ CE, _qui arcus_ CD, _duplus eſt. </
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<
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rũt_
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_enim rurſum triangula_ ACH,
<
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<
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, _æquiangula; </
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<
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_proinde, vt_ AC, _ad_ CH, _ita_ EC, _ad_
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: </
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<
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">_Et permutan-_
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">4. fexti.</
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_do, vt_ AC, _ad_ CE, _ita_ CH, _ad_
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. </
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<
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">_Cum ergo ſit, vt_
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AC, _ad_
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, _dimidium ipſius_ AC, _ad_ CH, _dimi-_
<
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<
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xlink:label
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">15. quinti.</
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_dium ipſius_
<
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; </
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<
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<
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">vt_
<
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, _ad_ CH, _ita_ CH, _ad_
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:
<
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</
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<
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">_proptereaq́;_ </
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<
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">CH, _ſinus rectus arcus_ CD, _medio loco propor-_
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_tionalis eſt inter_
<
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, _ſemiſſem ſinus totius,_ & </
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<
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, _ſinum ver_
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_ſum arcus_ CG, _qui duplus eſt arcus_ CD.</
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<
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<
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">HINC _fit, ſiperpendicularis_ EF, _ſemidiametrum_ AC, _ſecet bifariam, rectam_
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CH, _æqualem eſſe rectæ_
<
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>
. </
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>
<
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">_Si enim maior eſſet, aut minor, non poßet eſſe, vt_
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,
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_ad_ CH, _ita_ CH, _ad_
<
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: </
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<
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">_cum vna proportio eſſet maioris inæqualitatis, & </
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<
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_minoris inæqualitatis._</
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<
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">Sinus rectꝰ
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grad. 54. æ-
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qualis eſt
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ſemiſsi ſinꝰ
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totiꝰ, & ſi-
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nui gra. 18.
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ſimul. Sinꝰ
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aũt verſus
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grad. 72 æ-
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qualiseſt ſe
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miſſi ſinus
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totius, & ſi-
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nui verſo
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grad. 36. ſi-
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mul.</
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<
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<
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<
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tur ex ſemiſſe ſinus totius, & </
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<
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18. </
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<
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">Sinus autem verſus arcus grad. </
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<
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">72. </
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<
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">componitur
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ex ſemiſſe ſinus totius, & </
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<
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">ſinu verſo arcus grad. </
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<
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">36.</
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