Theodosius <Bithynius>; Clavius, Christoph
,
Theodosii Tripolitae Sphaericorum libri tres
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quadrata æqualia. </
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<
s
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xml:space
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">Cum ergo quadrata rectarum _EF, EG,_ æqualia quoque ſint, & </
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<
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xlink:label
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note-117-01
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xml:space
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">47. primi.</
note
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illi quidem æqualia ſint quadrata ex _FK, KE,_ huic vero quadrata ex _GL, LE;_ </
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<
s
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proinde duo quadrata ex _FK, KE,_ duobus quadratis ex _GL, LE,_ æqualia: </
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<
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xml:space
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">ſiau-
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ferantur duo æqualia quadrata rectarum _FK, GL,_ æqualia remanebunt quadra-
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ta ex _EK, EL;_ </
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<
s
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">ac proinde & </
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<
s
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xml:space
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">rectæ _EK, EL,_ æquales erunt Quare cum latera _EF,_
<
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_EK,_ lateribus _EG, EL,_ æqualia ſint, & </
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>
<
s
xml:id
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xml:space
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">baſis _FK,_ baſi _GL,_ æqualis; </
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>
<
s
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xml:space
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">erunt anguli
<
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<
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xlink:label
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xml:space
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">8. primi.</
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_FEB, GED,_ æquales; </
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<
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">ac proinde & </
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<
s
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">arcus _BF, DG,_ æquales erunt. </
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<
s
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xml:space
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">Quod ſi ſinus
<
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<
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xlink:label
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note-117-03a
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">26. tertij.</
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complementorum _EK, EL,_ ſint æquales, oſtendemus eodem modo, rectas _FK, GL,_
<
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æquales eſſe. </
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<
s
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xml:space
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">Quare vt prius, erunt arcus _BF, DG,_ æquales. </
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<
s
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xml:space
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">Si tandem ſinus verſi _KB,_
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_LD,_ ponantur æquales; </
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<
s
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xml:space
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">ijs ablatis ex ſemidiametris _EB, ED,_ relinquentur ſinus
<
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complementorum _EK, EL,_ æquales. </
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>
<
s
xml:id
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xml:space
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">Quare rurſus oſtendemus, vt prius, arcus _BF,_
<
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_DG,_ æquales eſſe. </
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<
s
xml:id
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xml:space
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">quoderat oſtendendum. </
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<
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xml:space
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">Iam vero ſit arcus _BF,_ maior arcu _DM,_
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& </
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<
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">ducatur ſinus _MO._ </
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<
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xml:space
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">Dico ſinum rectum _FK,_ maiorẽ eſſe ſinu recto _MO:_ </
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verſum _KB,_ maiorẽ ſinu verſo _OD:_ </
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xml:space
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">ſinũ vero cõplementi _EK,_ minorẽ ſinu cõplementi
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_EO._ </
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<
s
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xml:space
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">Poſito enim arcu _DG,_ æquali arcui _BF,_ erunt, vt demonſtr auimus, tam ſinus recti
<
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_FK,
<
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style
="
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>
L,_ quàm _OD,_ erit quoque ſinus verſus _KB,_ ſinu verſo _OD,_ maior: </
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>
<
s
xml:id
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xml:space
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">Item cũ _EL,_
<
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maior ſit, quàm _OD,_ erit quoque ſinus verſus _KB,_ ſinu verſo _OD,_ maior: </
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>
<
s
xml:id
="
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xml:space
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">Item cũ _EL,_
<
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minor ſit, quàm _EO,_ erit quoque ſinus cõplementi _EK,_ minor ſinu cõplementi _EO._
<
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/>
</
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<
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xml:space
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">Ducatur _MN,_ ad _GL,_ perpendicularis, eritque _NL,_ ipſi _MO,_ æqualis. </
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<
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xml:space
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">Cum ergo _GL,_
<
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<
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xlink:label
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">34. primi.</
note
>
maior ſit, quàm _NL,_ hoc eſt, quàm _MO,_ erit quoq; </
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<
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">ſinus rectus _FK,_ maior ſinu recto
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_MO._ </
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<
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">quod demonſt randum er at. </
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<
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xml:space
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">Sit denique tam ſinus rectus _FK,_ maior ſinu recto
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_MO,_ quàm ſinus verſus _KB,_ ſinu verſo _OD;_ </
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<
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">& </
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">ſinus complementi _EO,_ maior ſinu
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complementi _EK._ </
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<
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">Dico ſinui maiori tam recto, quàm verſo reſpondentem arcum _BF,_
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maiorem eſſe arcu _DM,_ qui minori ſinui tam recto, quàm verſo reſpondet. </
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<
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">At maio-
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ri ſinui complementi arcum reſpondentem _DM,_ minorem eſſe arcu _BF,_ qui minori
<
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ſinui complementi reſpondet. </
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<
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xml:space
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">Nam ſi _FK,_ maior ſit, quàm _MO,_ auferatur _KP,_ ipſi
<
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_MO,_ æqualis, & </
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<
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">ducatur _PQ,_ ad _FK,_ perpendicularis, ducaturque _QR,_ ad _BE,_
<
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perpendicularis, quæipſi _PK,_ hoc eſt, ipſi _MO,_ æqualis erit; </
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<
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">ac proinde, vt paulò ante
<
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<
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xlink:label
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note-117-05a
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">34. ptimi.</
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>
oſtenſum eſt, erunt arcus _BQ, DM,_ æquales, propter æqualitatem ſinuum rectorum
<
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_QR, MO._ </
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<
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xml:space
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">Cum ergoarcus _BF,_ arcu _BQ,_ maior ſit, erit idem arcus _BF,_ arcu _DM,_
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maior. </
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<
s
xml:id
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xml:space
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">Quòd ſi _KB,_ maior ſit, quàm _OD,_ abſcindatur _BR,_ ipſi _DO,_ æqualis, duca-
<
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turque _RQ,_ ad _BE,_ perpendicularis: </
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<
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">eruntq́ arcus _BQ, DM,_ vt paulo antemon-
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ſtrauimus, æquales, ob æqualitatem ſinuum verſorum _RB, OD._ </
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<
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xml:space
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">Quare cũ arcus _BF,_
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maior ſit arcu _
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Q,_ eritidem arcus _
<
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F,_ arcu _DM,_ maior. </
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<
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xml:space
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">Si tandem maior ſit _EO,_
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quàm _EK,_ detrahatur _EL,_ ipsi _
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K,_ æqualis, ducaturque ad _
<
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D,_ perpendicularis
<
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<
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">Anguli æ-
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quales ha-
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bent ſinus
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ęquales, &c.</
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_LG:_ </
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<
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">arcus _BF,
<
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,_ ob æqualitatem sinuum complementorum _
<
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">E</
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>
K,
<
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="
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L,_
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æquales, vt paulo ante fuit oſtenſum. </
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<
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xml:space
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">Quam ob rem cum arcus _DM,_ arcu _DG,_ sit
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minor, erit idem arcus _DM,_ arcn _BF,_ minor. </
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<
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">Quod eſt propositum.
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</
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</
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<
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<
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M_ prorſus dicendum eſt de sinubus angulorum. </
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<
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">Nam & </
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<
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">anguli æquales ha-
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<
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">Si in trian
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gulo rectã
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gulo latus
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recto angu
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lo oppoſitũ
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ſit ſinus to
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tus, erit
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vtrumuis
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laterum re
<
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liquorum
<
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ſinus rectꝰ
<
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anguli acu
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ti oppoſiti.</
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>
bent sinus æquales tam rectos, quam complemẽtorum, & </
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<
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<
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<
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æquales anguli insiſtunt in centro æqualibus arcubus, &</
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<
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">c.</
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<
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</
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<
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<
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MO_ in omni triangulo rectangulo, si latus recto angulo oppositum
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ponatur sinus totus, reliqua duo latera ſunt sinus recti reliquorum angulorum acu-
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torum, quibus opponuntur. </
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<
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">Vt in triangulo rectangulo _EKF,_ in quo _EF,_ eſt sinus to-
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tus, vtpote ſemidiameter circuli ex F, deſcripti, latus _
<
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>
K,_ eſtsinus rectus anguli
<
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_
<
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>
K,_ ex deſin. </
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<
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">6. </
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<
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">Sic quoque si idem circulus ex _
<
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="
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>
,_ deſcriberetur, eſſet latus _
<
emph
style
="
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">E</
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>
K,_ si-
<
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nus reclus anguli _
<
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style
="
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">EF</
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>
K,_ ex eadem deſin. </
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>
<
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">6. </
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>
<
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">Quod etiam hinc patet, quèd angulus _
<
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</
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