Theodosius <Bithynius>; Clavius, Christoph
,
Theodosii Tripolitae Sphaericorum libri tres
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<
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_HL_, _EK_, differentiæ inter ſinus _HI_, _EG_, & </
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>
<
s
xml:id
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xml:space
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<
s
xml:id
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xml:space
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">Quoniam vero arcus _DE_,
<
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ſi e xiguus fuerit, qualem nos ponimus, nempe Minutorum _45_. </
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>
<
s
xml:id
="
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xml:space
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">vel _15_. </
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>
<
s
xml:id
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"
xml:space
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">vel _5_. </
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>
<
s
xml:id
="
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"
xml:space
="
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">(Ex his
<
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/>
enim ſolis intermediorum arcuum ſinus inquirimus) à recta linea, quoad ſenſum, non
<
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<
figure
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fig-140-01
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xlink:href
="
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number
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<
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xlink:href
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/140-01
"/>
</
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>
differt, ac proinde multo minus arcus _DH_; </
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>
<
s
xml:id
="
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"
xml:space
="
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">erunt trianæ
<
lb
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gula _DEK_, _DHL_, quodammodo rectilinea. </
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>
<
s
xml:id
="
echoid-s5407
"
xml:space
="
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">& </
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>
<
s
xml:id
="
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"
xml:space
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">æquian-
<
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gula inter ſe. </
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<
s
xml:id
="
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xml:space
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preserve
">Quamobrem erit, vt _DE_, differentia inter
<
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<
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position
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xlink:label
="
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">4. fexti.</
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arcum _BD_, & </
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<
s
xml:id
="
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xml:space
="
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">arcum _BE_, ad _EK_, differentiam inter ſi-
<
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num _DF_, & </
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>
<
s
xml:id
="
echoid-s5411
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xml:space
="
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">ſinum _EG_, ita _DH_, differentia inter ar-
<
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cum _BD_, & </
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>
<
s
xml:id
="
echoid-s5412
"
xml:space
="
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">arcum _BH_, ad _HL_, differentiam inter ſi-
<
lb
/>
num _DF_, & </
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>
<
s
xml:id
="
echoid-s5413
"
xml:space
="
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">ſinum _HI_. </
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>
<
s
xml:id
="
echoid-s5414
"
xml:space
="
preserve
">Cum ergo priores tres magnitu-
<
lb
/>
dines ſint cognitæ, (nempe arcus _DE_, quo arcus _BE_, arcũ
<
lb
/>
_BD_, ſuperat; </
s
>
<
s
xml:id
="
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"
xml:space
="
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">& </
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>
<
s
xml:id
="
echoid-s5416
"
xml:space
="
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">recta _EK_, qua ſinus _EG_, ſinum _DF_,
<
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/>
excedit; </
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>
<
s
xml:id
="
echoid-s5417
"
xml:space
="
preserve
">nec non arcus _DH_, quo arcus _BH_, cuius ſinus
<
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/>
queritur, ſuperat arcum _BD_.) </
s
>
<
s
xml:id
="
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xml:space
="
preserve
">cognita etiam erit quarta magnitudo, id eſt, recta _HL_,
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quæ addenda eſt ſinui _DF_, vt fiat ſinus _HI_. </
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>
<
s
xml:id
="
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"
xml:space
="
preserve
">Conſtat igitur, ſinus per regulam pro-
<
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/>
portionum inuentos ſenſibiliter non differre à veris ſinubus, præſertim quando arcus
<
lb
/>
_DE_, quo arcus cognitorum ſinuuminter ſe differunt, valde exiguus eſt, ita vt à recta
<
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linea vix differat.</
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>
<
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</
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>
<
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<
s
xml:id
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xml:space
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">_MAGNVM_ quoque compendium in hoc negotio nobis afferet propoſitio octaua.
<
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</
s
>
<
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xml:id
="
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xml:space
="
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">
<
note
position
="
left
"
xlink:label
="
note-140-02
"
xlink:href
="
note-140-02a
"
xml:space
="
preserve
">Compédiũ
<
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miri ficum
<
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/>
pro inuen-
<
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/>
rione pluti
<
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/>
morum ſi-
<
lb
/>
nuum.</
note
>
Ex ea enim plurimos ſinus ex alijs inuentis per ſolam additionem, ſubtractionemue
<
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/>
nanciſcemur. </
s
>
<
s
xml:id
="
echoid-s5423
"
xml:space
="
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">_N_am ſi ſinum cuiuſuis arcus, qui maior non ſit, quàm grad. </
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>
<
s
xml:id
="
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"
xml:space
="
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">_30_. </
s
>
<
s
xml:id
="
echoid-s5425
"
xml:space
="
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">addamus
<
lb
/>
ſinui arcus, qui ab arcugrad. </
s
>
<
s
xml:id
="
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"
xml:space
="
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">_60_. </
s
>
<
s
xml:id
="
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"
xml:space
="
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">ſuperatur ſumpto illo arcu, componemus ſinum arcus
<
lb
/>
qui eodemillo arcu aſſumpto arcũgrad. </
s
>
<
s
xml:id
="
echoid-s5428
"
xml:space
="
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">_30_. </
s
>
<
s
xml:id
="
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"
xml:space
="
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">ſuperat: </
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>
<
s
xml:id
="
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"
xml:space
="
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">propterea quod differentia inter
<
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/>
ſinus duorum horum arcuum maiorum æqualis eſt ſinui arcus illius aſſumpti, qui ma
<
lb
/>
ior nõ ponitur, quàm grad. </
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>
<
s
xml:id
="
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"
xml:space
="
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">_30_. </
s
>
<
s
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="
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"
xml:space
="
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">vt ibi demonſtrauimus. </
s
>
<
s
xml:id
="
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"
xml:space
="
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">Vt ſi _3461171_. </
s
>
<
s
xml:id
="
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"
xml:space
="
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">ſinũ arcus grad.
<
lb
/>
</
s
>
<
s
xml:id
="
echoid-s5435
"
xml:space
="
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">_20_. </
s
>
<
s
xml:id
="
echoid-s5436
"
xml:space
="
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">Min. </
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>
<
s
xml:id
="
echoid-s5437
"
xml:space
="
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">_15_. </
s
>
<
s
xml:id
="
echoid-s5438
"
xml:space
="
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">adijciamus ad _6394390_. </
s
>
<
s
xml:id
="
echoid-s5439
"
xml:space
="
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">ſinum arcus grad. </
s
>
<
s
xml:id
="
echoid-s5440
"
xml:space
="
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">_39_. </
s
>
<
s
xml:id
="
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xml:space
="
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">Min. </
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>
<
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="
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xml:space
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">_45._</
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>
<
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="
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xml:space
="
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">. </
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>
<
s
xml:id
="
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"
xml:space
="
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">qui abarcu grad. </
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>
<
s
xml:id
="
echoid-s5445
"
xml:space
="
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">
<
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/>
_60_. </
s
>
<
s
xml:id
="
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"
xml:space
="
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">ſuperatur dicto arcu grad. </
s
>
<
s
xml:id
="
echoid-s5447
"
xml:space
="
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">_20_. </
s
>
<
s
xml:id
="
echoid-s5448
"
xml:space
="
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">Min. </
s
>
<
s
xml:id
="
echoid-s5449
"
xml:space
="
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">_15_. </
s
>
<
s
xml:id
="
echoid-s5450
"
xml:space
="
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">conficiemus _9855561_. </
s
>
<
s
xml:id
="
echoid-s5451
"
xml:space
="
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">ſinum arcus grad. </
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>
<
s
xml:id
="
echoid-s5452
"
xml:space
="
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">
<
lb
/>
_80_. </
s
>
<
s
xml:id
="
echoid-s5453
"
xml:space
="
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">Min. </
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>
<
s
xml:id
="
echoid-s5454
"
xml:space
="
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">_15_. </
s
>
<
s
xml:id
="
echoid-s5455
"
xml:space
="
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">qui arcum gra. </
s
>
<
s
xml:id
="
echoid-s5456
"
xml:space
="
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">_60_. </
s
>
<
s
xml:id
="
echoid-s5457
"
xml:space
="
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">eodem arcu grad. </
s
>
<
s
xml:id
="
echoid-s5458
"
xml:space
="
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">_20_. </
s
>
<
s
xml:id
="
echoid-s5459
"
xml:space
="
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">Min _15_. </
s
>
<
s
xml:id
="
echoid-s5460
"
xml:space
="
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">ſuperat. </
s
>
<
s
xml:id
="
echoid-s5461
"
xml:space
="
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">Sic ſi _5000000_. </
s
>
<
s
xml:id
="
echoid-s5462
"
xml:space
="
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">
<
lb
/>
ſinum arcus grad. </
s
>
<
s
xml:id
="
echoid-s5463
"
xml:space
="
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">_30_. </
s
>
<
s
xml:id
="
echoid-s5464
"
xml:space
="
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">addamus ſinui _5000000_. </
s
>
<
s
xml:id
="
echoid-s5465
"
xml:space
="
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">arcus grad _30_. </
s
>
<
s
xml:id
="
echoid-s5466
"
xml:space
="
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">quem arcus grad. </
s
>
<
s
xml:id
="
echoid-s5467
"
xml:space
="
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">_60_. </
s
>
<
s
xml:id
="
echoid-s5468
"
xml:space
="
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">
<
lb
/>
fuperat dicto illo arcu grad. </
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>
<
s
xml:id
="
echoid-s5469
"
xml:space
="
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">_30_. </
s
>
<
s
xml:id
="
echoid-s5470
"
xml:space
="
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">componemus _10000000_. </
s
>
<
s
xml:id
="
echoid-s5471
"
xml:space
="
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">ſinum arcus grad. </
s
>
<
s
xml:id
="
echoid-s5472
"
xml:space
="
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">_90_. </
s
>
<
s
xml:id
="
echoid-s5473
"
xml:space
="
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">qui
<
lb
/>
arcum grad. </
s
>
<
s
xml:id
="
echoid-s5474
"
xml:space
="
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">_60_. </
s
>
<
s
xml:id
="
echoid-s5475
"
xml:space
="
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">eodem illo arcu aſſumpto grad. </
s
>
<
s
xml:id
="
echoid-s5476
"
xml:space
="
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">_30_. </
s
>
<
s
xml:id
="
echoid-s5477
"
xml:space
="
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">ſuperat.</
s
>
<
s
xml:id
="
echoid-s5478
"
xml:space
="
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"/>
</
p
>
<
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style
="
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">
<
s
xml:id
="
echoid-s5479
"
xml:space
="
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">_ITEM_ ſi ſinum cuiuslibet arcus, qui arcu grad. </
s
>
<
s
xml:id
="
echoid-s5480
"
xml:space
="
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">_30_. </
s
>
<
s
xml:id
="
echoid-s5481
"
xml:space
="
preserve
">maior non ſit, ſubducamus ex
<
lb
/>
ſinu arcus, qui arcum grad _60_. </
s
>
<
s
xml:id
="
echoid-s5482
"
xml:space
="
preserve
">ſumpto illo arcu ſuperat, relinquetur ſinus arcus, qui
<
lb
/>
eodem illo arcu aſſumpto ab arcu grad. </
s
>
<
s
xml:id
="
echoid-s5483
"
xml:space
="
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">_60_. </
s
>
<
s
xml:id
="
echoid-s5484
"
xml:space
="
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">ſuperatur. </
s
>
<
s
xml:id
="
echoid-s5485
"
xml:space
="
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">Vt ſi _3502075_. </
s
>
<
s
xml:id
="
echoid-s5486
"
xml:space
="
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">ſinum arcus
<
lb
/>
grad. </
s
>
<
s
xml:id
="
echoid-s5487
"
xml:space
="
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">_20_. </
s
>
<
s
xml:id
="
echoid-s5488
"
xml:space
="
preserve
">Min. </
s
>
<
s
xml:id
="
echoid-s5489
"
xml:space
="
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">_30_ detrahamus ex _9862856_. </
s
>
<
s
xml:id
="
echoid-s5490
"
xml:space
="
preserve
">ſinu arcus grad _80_. </
s
>
<
s
xml:id
="
echoid-s5491
"
xml:space
="
preserve
">Min. </
s
>
<
s
xml:id
="
echoid-s5492
"
xml:space
="
preserve
">_30_. </
s
>
<
s
xml:id
="
echoid-s5493
"
xml:space
="
preserve
">qui arcum
<
lb
/>
grad. </
s
>
<
s
xml:id
="
echoid-s5494
"
xml:space
="
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">_60_. </
s
>
<
s
xml:id
="
echoid-s5495
"
xml:space
="
preserve
">dicto arcu grad. </
s
>
<
s
xml:id
="
echoid-s5496
"
xml:space
="
preserve
">_20_. </
s
>
<
s
xml:id
="
echoid-s5497
"
xml:space
="
preserve
">Min. </
s
>
<
s
xml:id
="
echoid-s5498
"
xml:space
="
preserve
">_30_. </
s
>
<
s
xml:id
="
echoid-s5499
"
xml:space
="
preserve
">ſuperat, reliquus erit _6360781_. </
s
>
<
s
xml:id
="
echoid-s5500
"
xml:space
="
preserve
">ſinus arcus gra.
<
lb
/>
</
s
>
<
s
xml:id
="
echoid-s5501
"
xml:space
="
preserve
">_39_. </
s
>
<
s
xml:id
="
echoid-s5502
"
xml:space
="
preserve
">Min. </
s
>
<
s
xml:id
="
echoid-s5503
"
xml:space
="
preserve
">_30_. </
s
>
<
s
xml:id
="
echoid-s5504
"
xml:space
="
preserve
">quem arcus grad. </
s
>
<
s
xml:id
="
echoid-s5505
"
xml:space
="
preserve
">_60_. </
s
>
<
s
xml:id
="
echoid-s5506
"
xml:space
="
preserve
">eodemillo arcu grad. </
s
>
<
s
xml:id
="
echoid-s5507
"
xml:space
="
preserve
">_20_. </
s
>
<
s
xml:id
="
echoid-s5508
"
xml:space
="
preserve
">Min. </
s
>
<
s
xml:id
="
echoid-s5509
"
xml:space
="
preserve
">_30_ ſuperat.</
s
>
<
s
xml:id
="
echoid-s5510
"
xml:space
="
preserve
"/>
</
p
>
<
p
style
="
it
">
<
s
xml:id
="
echoid-s5511
"
xml:space
="
preserve
">_RVRSVS_, ſi ex ſinu cuiuſuis arcus, qui maior ſit arcu grad. </
s
>
<
s
xml:id
="
echoid-s5512
"
xml:space
="
preserve
">_60_. </
s
>
<
s
xml:id
="
echoid-s5513
"
xml:space
="
preserve
">detrahatur ſinus
<
lb
/>
arcus, quitanto minor ſit arcu grad. </
s
>
<
s
xml:id
="
echoid-s5514
"
xml:space
="
preserve
">_60_. </
s
>
<
s
xml:id
="
echoid-s5515
"
xml:space
="
preserve
">quanto ille maior eſt, relinquetur ſinus ar-
<
lb
/>
cus, quo vteruis illorũ ab arcu grad. </
s
>
<
s
xml:id
="
echoid-s5516
"
xml:space
="
preserve
">_60_. </
s
>
<
s
xml:id
="
echoid-s5517
"
xml:space
="
preserve
">differt. </
s
>
<
s
xml:id
="
echoid-s5518
"
xml:space
="
preserve
">Vt ſi ex _9781476_. </
s
>
<
s
xml:id
="
echoid-s5519
"
xml:space
="
preserve
">ſinu arcus gra. </
s
>
<
s
xml:id
="
echoid-s5520
"
xml:space
="
preserve
">_78_.
<
lb
/>
</
s
>
<
s
xml:id
="
echoid-s5521
"
xml:space
="
preserve
">auferatur _6691306_. </
s
>
<
s
xml:id
="
echoid-s5522
"
xml:space
="
preserve
">ſinus arcus gra. </
s
>
<
s
xml:id
="
echoid-s5523
"
xml:space
="
preserve
">_42_. </
s
>
<
s
xml:id
="
echoid-s5524
"
xml:space
="
preserve
">reliquus erit _3090170_. </
s
>
<
s
xml:id
="
echoid-s5525
"
xml:space
="
preserve
">ſinus arcus grad. </
s
>
<
s
xml:id
="
echoid-s5526
"
xml:space
="
preserve
">_18_. </
s
>
<
s
xml:id
="
echoid-s5527
"
xml:space
="
preserve
">
<
lb
/>
quo vterq; </
s
>
<
s
xml:id
="
echoid-s5528
"
xml:space
="
preserve
">illorũ ab arcu grad. </
s
>
<
s
xml:id
="
echoid-s5529
"
xml:space
="
preserve
">_60_. </
s
>
<
s
xml:id
="
echoid-s5530
"
xml:space
="
preserve
">differt. </
s
>
<
s
xml:id
="
echoid-s5531
"
xml:space
="
preserve
">Quæ omnia ex dicta propeſ. </
s
>
<
s
xml:id
="
echoid-s5532
"
xml:space
="
preserve
">_8_. </
s
>
<
s
xml:id
="
echoid-s5533
"
xml:space
="
preserve
">colliguntur.</
s
>
<
s
xml:id
="
echoid-s5534
"
xml:space
="
preserve
"/>
</
p
>
<
p
style
="
it
">
<
s
xml:id
="
echoid-s5535
"
xml:space
="
preserve
">_ITAQVE_ ſatis eſt, vt inueniantur per regulam proportionum ſinus ommum ar-
<
lb
/>
cuum à principio quadrantis vſque ad arcum grad. </
s
>
<
s
xml:id
="
echoid-s5536
"
xml:space
="
preserve
">_30_. </
s
>
<
s
xml:id
="
echoid-s5537
"
xml:space
="
preserve
">Si enim ex his eliciantur ſinus
<
lb
/>
complementorum, & </
s
>
<
s
xml:id
="
echoid-s5538
"
xml:space
="
preserve
">ex his inuentis detrahãtur priores illi ſinus, reli qui erunt ſinus
<
lb
/>
omnium arcuum inter arcum grad. </
s
>
<
s
xml:id
="
echoid-s5539
"
xml:space
="
preserve
">_30_ & </
s
>
<
s
xml:id
="
echoid-s5540
"
xml:space
="
preserve
">arcum grad. </
s
>
<
s
xml:id
="
echoid-s5541
"
xml:space
="
preserve
">_90_. </
s
>
<
s
xml:id
="
echoid-s5542
"
xml:space
="
preserve
">Item ſicogniti eſſent ſinus
<
lb
/>
arcuũ omniũ ab arcu grad. </
s
>
<
s
xml:id
="
echoid-s5543
"
xml:space
="
preserve
">_30_. </
s
>
<
s
xml:id
="
echoid-s5544
"
xml:space
="
preserve
">vſq; </
s
>
<
s
xml:id
="
echoid-s5545
"
xml:space
="
preserve
">ad finem quadrantis, & </
s
>
<
s
xml:id
="
echoid-s5546
"
xml:space
="
preserve
">ſinus omniũ arcuum, </
s
>
</
p
>
</
div
>
</
text
>
</
echo
>