Theodosius <Bithynius>; Clavius, Christoph
,
Theodosii Tripolitae Sphaericorum libri tres
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<
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">SINT iam in eodem triangulo ABC, duo arcus AB, AC, quadrante
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quidem maiores, at BC, quadrans. </
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<
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xml:space
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">Autigitur arcus AB, AC, æquales ſunt,
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aut inæquales. </
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<
s
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xml:space
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">Si æquales, erunt duo anguli B, C, obtuſi. </
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<
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">Sit quadrans BD,
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fig-396-01
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236
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396-01
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xlink:href
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note-396-01
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xml:space
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">25. huius.</
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& </
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<
s
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xml:space
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">per puncta C, D, arcus CD, maximi circuli du-
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<
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position
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xlink:label
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note-396-02
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xml:space
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">20. 1 Theod.</
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catur conueniens cum arcu CA, protracto in E.
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</
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<
s
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xml:space
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">Quia igitur arcus BC, BD, quadrantes ſunt, erũt
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anguli D, & </
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<
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">BCD, recti; </
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<
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">& </
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<
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xml:space
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">arcus CD, propter
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<
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xlink:label
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note-396-03
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">25. huius</
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angulum B, quem obtuſum eſſe oſtendimus, qua-
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drante maior: </
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<
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<
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">arcus AC, qua-
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<
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xlink:label
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note-396-04
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">26. huius.</
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drante maior. </
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<
s
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xml:space
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">Igitur arcus CD, CA, ſimul ma-
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iores ſunt ſemicirculo; </
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>
<
s
xml:id
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xml:space
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">ac propterea, cum arcus
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CDE, CAE, circulum conficiant, (quòd vter-
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que ſemicirculus ſit.) </
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<
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">erunt arcus ED, EA, ſemi-
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<
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circulo minores. </
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<
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">Quare angulus EDB, qui rectus
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eſt, (quòd duo anguli ad D, æquales ſint duobus
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">5. huius.</
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rectis, & </
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<
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">angulus BDC, oſtenſus ſit rectus.) </
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<
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">maior
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erit angulo EAD; </
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<
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xml:space
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">atque adeo EAD, acutus erit.
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</
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<
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<
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xlink:label
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xml:space
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">14. huius.</
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Cum ergo anguli EAD, DAC, duobus rectis ſint
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<
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xlink:label
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">5. huius.</
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æquales, erit BAC, obtuſus. </
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<
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">Sunt etiam anguli B, C, demonſtrati obtuſi.
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</
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<
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xml:space
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">Tres igitur anguli A, B, C, trianguli ABC, obtuſi ſunt.</
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<
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<
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">SI verò AB, AC, latera, quæ quadrante maiora ſunt, non ſunt æqualia,
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ſit maius AC; </
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<
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">& </
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<
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">abſcindatur arcus AD, æqualis arcui AB; </
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xml:space
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">& </
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<
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xml:space
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">per puncta B,
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">1. huius.</
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D, tranſeat arcus BD, circuli maximi: </
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">eritq́ue adhuc arcus AD, maior qua-
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<
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xlink:href
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<
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xlink:label
="
note-396-10
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xlink:href
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xml:space
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">20. 1 Theod.</
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drante, cum ei æqualis AB, maior etiam ponatur-
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Anguli igitur ADB, ABD, obtuſi ſunt. </
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<
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">Multo
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<
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">25. huius.</
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ergo magis obtuſus erit angulus ABC. </
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<
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">Sit qua-
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drans BE, & </
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<
s
xml:id
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">per puncta C, E, tranſeat arcus CE,
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<
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position
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xlink:label
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xlink:href
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">20. 1 Theod.</
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circuli maximi occurrens arcui CA, producto in
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F. </
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<
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">Quoniam igitur quadrantes ſunt BE, BC, & </
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<
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angulus EBC, oſtẽſus eſt obtuſus, erit arcus EC,
<
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<
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position
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xlink:label
="
note-396-13
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xlink:href
="
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xml:space
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">26. huius.</
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maior quadrãte, ſed anguli E, & </
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<
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xml:id
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">BCE, recti erunt.
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</
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<
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<
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xlink:label
="
note-396-14
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xlink:href
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">25. huius.</
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Angulus ergo ACB, obtuſus erit. </
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<
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xml:id
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xml:space
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">Et quoniam
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arcus CE, oſtenſus eſt quadrante maior, & </
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<
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xml:id
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">arcus
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AC, maior etiam ponitur, quam quadrans; </
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>
<
s
xml:id
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xml:space
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">erunt
<
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arcus CE, CA, ſimul ſemicirculo maiores. </
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<
s
xml:id
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">Cũ ergo
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arcus CEF, CAF, integro circulo æquales ſint,
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<
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xlink:label
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xml:space
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">21. 1. Theod.</
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quòd vterque ſit ſemicirculus, erũt arcus FE, FA,
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ſimul ſemicirculo minores. </
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>
<
s
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">Quamobrem angulus
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FEB, quirectus eſt, (ſunt enim duo anguli ad E, duobus rectis æquales, & </
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<
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<
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xlink:label
="
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">5. huius.</
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angulus BEC, oſtenſus eſt rectus.) </
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<
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xml:id
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">maior erit angulo FAE. </
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<
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xml:id
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">Acutus ergo eſt
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<
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">14. huius.</
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angulus FAE; </
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<
s
xml:id
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xml:space
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">ac propterea, cum duo anguli ad A, ſint æquales duobus rectis,
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<
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xlink:label
="
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xlink:href
="
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xml:space
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">5. huius.</
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angulus BAc, obtuſus erit. </
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<
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xml:id
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xml:space
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">Sunt autem etiam oſtenſi obtuſi anguli ABC,
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ACB. </
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<
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xml:id
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xml:space
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">Tres igitur anguli in triangulo ABC, obtuſi ſunt. </
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>
<
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xml:id
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">In omni ergo trian
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gulo ſphærico, cuius omnes arcus, &</
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<
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">c. </
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<
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">Quod erat demonſtrandum.</
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<
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<
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">Non enim omne triangulum ſphæricum, cu-
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ius omnes anguli ſunt obtuſi, neceſſario habet omnes arcus quadrante maiores, </
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