Theodosius <Bithynius>; Clavius, Christoph - Theodosii Tripolitae Sphaericorum libri tres

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140128 _HL_, _EK_, differentiæ inter ſinus _HI_, _EG_, & ſinum _DT_. Quoniam vero arcus _DE_,
ſi e xiguus fuerit, qualem nos ponimus, nempe Minutorum _45_. vel _15_. vel _5_. (Ex his
enim ſolis intermediorum arcuum ſinus inquirimus) à recta linea, quoad ſenſum, non

133[Figure 133] differt, ac proinde multo minus arcus _DH_; erunt trianæ
gula _DEK_, _DHL_, quodammodo rectilinea. & æquian-
gula inter ſe. Quamobrem erit, vt _DE_, differentia inter
114. fexti. arcum _BD_, & arcum _BE_, ad _EK_, differentiam inter ſi-
num _DF_, & ſinum _EG_, ita _DH_, differentia inter ar-
cum _BD_, & arcum _BH_, ad _HL_, differentiam inter ſi-
num _DF_, & ſinum _HI_. Cum ergo priores tres magnitu-
dines ſint cognitæ, (nempe arcus _DE_, quo arcus _BE_, arcũ
_BD_, ſuperat; & recta _EK_, qua ſinus _EG_, ſinum _DF_,
excedit; nec non arcus _DH_, quo arcus _BH_, cuius ſinus
queritur, ſuperat arcum _BD_.) cognita etiam erit quarta magnitudo, id eſt, recta _HL_,
quæ addenda eſt ſinui _DF_, vt fiat ſinus _HI_. Conſtat igitur, ſinus per regulam pro-
portionum inuentos ſenſibiliter non differre à veris ſinubus, præſertim quando arcus
_DE_, quo arcus cognitorum ſinuuminter ſe differunt, valde exiguus eſt, ita vt à recta
linea vix differat.

_MAGNVM_ quoque compendium in hoc negotio nobis afferet propoſitio octaua.
22Compédiũ
miri ficum
pro inuen-
rione pluti
morum ſi-
nuum. Ex ea enim plurimos ſinus ex alijs inuentis per ſolam additionem, ſubtractionemue
nanciſcemur. _N_am ſi ſinum cuiuſuis arcus, qui maior non ſit, quàm grad. _30_. addamus
ſinui arcus, qui ab arcugrad. _60_. ſuperatur ſumpto illo arcu, componemus ſinum arcus
qui eodemillo arcu aſſumpto arcũgrad. _30_. ſuperat: propterea quod differentia inter
ſinus duorum horum arcuum maiorum æqualis eſt ſinui arcus illius aſſumpti, qui ma
ior nõ ponitur, quàm grad. _30_. vt ibi demonſtrauimus. Vt ſi _3461171_. ſinũ arcus grad.
_20_. Min. _15_. adĳciamus ad _6394390_. ſinum arcus grad. _39_. Min. _45._ . qui abarcu grad.
_60_. ſuperatur dicto arcu grad. _20_. Min. _15_. conficiemus _9855561_. ſinum arcus grad.
_80_. Min. _15_. qui arcum gra. _60_. eodem arcu grad. _20_. Min _15_. ſuperat. Sic ſi _5000000_.
ſinum arcus grad. _30_. addamus ſinui _5000000_. arcus grad _30_. quem arcus grad. _60_.
fuperat dicto illo arcu grad. _30_. componemus _10000000_. ſinum arcus grad. _90_. qui
arcum grad. _60_. eodem illo arcu aſſumpto grad. _30_. ſuperat.

_ITEM_ ſi ſinum cuiuslibet arcus, qui arcu grad. _30_. maior non ſit, ſubducamus ex
ſinu arcus, qui arcum grad _60_. ſumpto illo arcu ſuperat, relinquetur ſinus arcus, qui
eodem illo arcu aſſumpto ab arcu grad. _60_. ſuperatur. Vt ſi _3502075_. ſinum arcus
grad. _20_. Min. _30_ detrahamus ex _9862856_. ſinu arcus grad _80_. Min. _30_. qui arcum
grad. _60_. dicto arcu grad. _20_. Min. _30_. ſuperat, reliquus erit _6360781_. ſinus arcus gra.
_39_. Min. _30_. quem arcus grad. _60_. eodemillo arcu grad. _20_. Min. _30_ ſuperat.

_RVRSVS_, ſi ex ſinu cuiuſuis arcus, qui maior ſit arcu grad. _60_. detrahatur ſinus
arcus, quitanto minor ſit arcu grad. _60_. quanto ille maior eſt, relinquetur ſinus ar-
cus, quo vteruis illorũ ab arcu grad. _60_. differt. Vt ſi ex _9781476_. ſinu arcus gra. _78_.
auferatur _6691306_. ſinus arcus gra. _42_. reliquus erit _3090170_. ſinus arcus grad. _18_.
quo vterq; illorũ ab arcu grad. _60_. differt. Quæ omnia ex dicta propeſ. _8_. colliguntur.

_ITAQVE_ ſatis eſt, vt inueniantur per regulam proportionum ſinus ommum ar-
cuum à principio quadrantis vſque ad arcum grad. _30_. Si enim ex his eliciantur ſinus
complementorum, & ex his inuentis detrahãtur priores illi ſinus, reli qui erunt ſinus
omnium arcuum inter arcum grad. _30_ & arcum grad. _90_. Item ſicogniti eſſent ſinus
arcuũ omniũ ab arcu grad. _30_. vſq; ad finem quadrantis, & ſinus omniũ arcuum,

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