Clavius, Christoph
,
Gnomonices libri octo, in quibus non solum horologiorum solariu[m], sed aliarum quo[quam] rerum, quae ex gnomonis umbra cognosci possunt, descriptiones geometricè demonstrantur
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GNOMONICES
"/>
eodem puncto F. </
s
>
<
s
xml:id
="
echoid-s3880
"
xml:space
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">Si enim puncta cont actuum C, D, per diametrum ſunt oppoſita, ita vt arcus C G, D G,
<
lb
/>
ſint quadrantes, perſpicuum eſt ex ijs, quæ proximè demonſtrata ſunt, tres hos circulos ſe mutuo interſe-
<
lb
/>
care in Aequatore in vno eodemq́, puncto. </
s
>
<
s
xml:id
="
echoid-s3881
"
xml:space
="
preserve
">Si verò puncta contactuum C, D, non ſunt oppoſita, deſcri-
<
lb
/>
bantur per polum E, & </
s
>
<
s
xml:id
="
echoid-s3882
"
xml:space
="
preserve
">per contactus C, D, circuli maximi E C, E D. </
s
>
<
s
xml:id
="
echoid-s3883
"
xml:space
="
preserve
">Item per puncta C, G, arcus
<
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/>
<
figure
xlink:label
="
fig-0080-01
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xlink:href
="
fig-0080-01a
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number
="
61
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<
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file
="
0080-01
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xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0080-01
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</
figure
>
circuli maximi C H G, & </
s
>
<
s
xml:id
="
echoid-s3884
"
xml:space
="
preserve
">per puncta G, D, arcus
<
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/>
maximi circuli G I D, deſcribatur, ducantur{q́ue} chor
<
lb
/>
dę C G, G D. </
s
>
<
s
xml:id
="
echoid-s3885
"
xml:space
="
preserve
">Quoniam igitur per defin. </
s
>
<
s
xml:id
="
echoid-s3886
"
xml:space
="
preserve
">poli à Theo-
<
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/>
doſio traditam, rectæ ex polo E, ad puncta C, D,
<
lb
/>
cadentes æquales ſunt, erũt & </
s
>
<
s
xml:id
="
echoid-s3887
"
xml:space
="
preserve
">arcus E C, E D, æqua-
<
lb
/>
<
note
position
="
left
"
xlink:label
="
note-0080-01
"
xlink:href
="
note-0080-01a
"
xml:space
="
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">28. tertij.</
note
>
les. </
s
>
<
s
xml:id
="
echoid-s3888
"
xml:space
="
preserve
">Rurſus, quia arcus C G, G D, paralleli C D,
<
lb
/>
<
note
position
="
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"
xlink:label
="
note-0080-02
"
xlink:href
="
note-0080-02a
"
xml:space
="
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">10</
note
>
æquales ſunt, erunt & </
s
>
<
s
xml:id
="
echoid-s3889
"
xml:space
="
preserve
">rectę C G, G D, æquales. </
s
>
<
s
xml:id
="
echoid-s3890
"
xml:space
="
preserve
">Igi-
<
lb
/>
<
note
position
="
left
"
xlink:label
="
note-0080-03
"
xlink:href
="
note-0080-03a
"
xml:space
="
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">29. tertij.</
note
>
tur & </
s
>
<
s
xml:id
="
echoid-s3891
"
xml:space
="
preserve
">arcus maximorum circulorum C H G, G I D,
<
lb
/>
<
note
position
="
left
"
xlink:label
="
note-0080-04
"
xlink:href
="
note-0080-04a
"
xml:space
="
preserve
">28. tertij.</
note
>
<
note
position
="
left
"
xlink:label
="
note-0080-05
"
xlink:href
="
note-0080-05a
"
xml:space
="
preserve
">Duo circuli ho
<
lb
/>
rarum ab or.
<
lb
/>
vel ò
<
unsure
/>
cc. tangen-
<
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/>
tes maximum
<
lb
/>
parallelorũ ſem
<
lb
/>
per apparentiũ
<
lb
/>
in duob
<
unsure
/>
us pun-
<
lb
/>
ctis quibuſcun-
<
lb
/>
que, & circulus
<
lb
/>
horæ à mer. uel
<
lb
/>
med. noc. ſecans
<
lb
/>
eundem paral-
<
lb
/>
lela
<
unsure
/>
m in pun-
<
lb
/>
cto æqualiter à
<
lb
/>
punctis conta-
<
lb
/>
ctuum diſtante,
<
lb
/>
ſe mutuo ſecãt
<
lb
/>
in vno eodéq́ue
<
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/>
puncto.</
note
>
ęquales erunt. </
s
>
<
s
xml:id
="
echoid-s3892
"
xml:space
="
preserve
">Quare duo latera E C, E G, triangu-
<
lb
/>
li ſphærici C E G, duobus lateribus E D, E G, trian-
<
lb
/>
guli ſphęrici D E G, ęqualia erunt: </
s
>
<
s
xml:id
="
echoid-s3893
"
xml:space
="
preserve
">Sunt autem & </
s
>
<
s
xml:id
="
echoid-s3894
"
xml:space
="
preserve
">
<
lb
/>
baſes C H G, D I G, æquales. </
s
>
<
s
xml:id
="
echoid-s3895
"
xml:space
="
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">Igitur per propoſ. </
s
>
<
s
xml:id
="
echoid-s3896
"
xml:space
="
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">35.
<
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/>
</
s
>
<
s
xml:id
="
echoid-s3897
"
xml:space
="
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">lib. </
s
>
<
s
xml:id
="
echoid-s3898
"
xml:space
="
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">3. </
s
>
<
s
xml:id
="
echoid-s3899
"
xml:space
="
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">Ioan. </
s
>
<
s
xml:id
="
echoid-s3900
"
xml:space
="
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">Regiom. </
s
>
<
s
xml:id
="
echoid-s3901
"
xml:space
="
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">de triangulis, vel per propoſ. </
s
>
<
s
xml:id
="
echoid-s3902
"
xml:space
="
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">
<
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/>
18. </
s
>
<
s
xml:id
="
echoid-s3903
"
xml:space
="
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">nostrorum triangulorum ſphęricorum, anguli
<
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/>
C E G, D E G, ęquales erunt, ac proinde arcus E G, angulum C E D, diuidet bifariam. </
s
>
<
s
xml:id
="
echoid-s3904
"
xml:space
="
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">Quoniam verò
<
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/>
circulus maximus per polum E, & </
s
>
<
s
xml:id
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"
xml:space
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">punctum F, deſcriptus diuidit eundem angulum CED, bifariam, vt
<
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<
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position
="
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xlink:label
="
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"
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="
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"
xml:space
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">20</
note
>
mox demonſtrabimus, perſpicuum eſt, circulum maximum E G, productum per punctum F, tranſire:
<
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</
s
>
<
s
xml:id
="
echoid-s3906
"
xml:space
="
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">alias duo maximi circuli, nempe E G, & </
s
>
<
s
xml:id
="
echoid-s3907
"
xml:space
="
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">ille qui ex E, per F, ducitur, diuiderent eundem angulum
<
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/>
C E D, bifariam, quod est
<
unsure
/>
abſurdum. </
s
>
<
s
xml:id
="
echoid-s3908
"
xml:space
="
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">Quòd autem circulus maximus per E, & </
s
>
<
s
xml:id
="
echoid-s3909
"
xml:space
="
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">F, deſcriptus di-
<
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uidat bifariam angulum C E D, ita demonſtrabimus. </
s
>
<
s
xml:id
="
echoid-s3910
"
xml:space
="
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">Intelligatur per polum E, & </
s
>
<
s
xml:id
="
echoid-s3911
"
xml:space
="
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">per F, deſcri-
<
lb
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ptus circulus maximus E F H. </
s
>
<
s
xml:id
="
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"
xml:space
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">Dico angulum C E F, æqualem eſſe angulo D E F. </
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>
<
s
xml:id
="
echoid-s3913
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xml:space
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">Quoniam enim cir-
<
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culi maximi E C, E D, tranſeunt, per propoſ. </
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<
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xml:space
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">5. </
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<
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<
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">2. </
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>
<
s
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xml:space
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">Theodoſii, per polos circulorum C F, D F, quòd per
<
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contactus C, D, & </
s
>
<
s
xml:id
="
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"
xml:space
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">per polum E, circuli C D, ducti ſint; </
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>
<
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="
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xml:space
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">Tranſeunt autem & </
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<
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xml:id
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xml:space
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A B; </
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<
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xml:id
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<
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xml:id
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xml:space
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">9. </
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>
<
s
xml:id
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echoid-s3923
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xml:space
="
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">lib. </
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>
<
s
xml:id
="
echoid-s3924
"
xml:space
="
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">2. </
s
>
<
s
xml:id
="
echoid-s3925
"
xml:space
="
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">Theodoſii, circulus E C, ſecet ſegmenta circulorum C B, A B, ſe ſe in
<
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B, ſe
<
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/>
cantium, quæ quidem, per propoſ. </
s
>
<
s
xml:id
="
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xml:space
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">11. </
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>
<
s
xml:id
="
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"
xml:space
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">lib. </
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>
<
s
xml:id
="
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"
xml:space
="
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">1. </
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>
<
s
xml:id
="
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"
xml:space
="
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">Theodoſii, ſemicirculi ſunt, bifariam: </
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>
<
s
xml:id
="
echoid-s3930
"
xml:space
="
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">Ac propterea ar-
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cus C B, ſit quadrans. </
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>
<
s
xml:id
="
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"
xml:space
="
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">Eadem{q́ue} ratione quadrans erit arcus D A. </
s
>
<
s
xml:id
="
echoid-s3932
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xml:space
="
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">Quia vero per Theorema 1. </
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>
<
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xml:id
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xml:space
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">ſcholij
<
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<
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xlink:label
="
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="
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xml:space
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">30</
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>
propoſ. </
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>
<
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="
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xml:space
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">21. </
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>
<
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xml:id
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">lib. </
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<
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xml:space
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">2. </
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>
<
s
xml:id
="
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xml:space
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">Theodoſii, circuli maximi C B, D N, eundem parallelum C D, tangentes, æqualiter in-
<
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clinantur ad A B, maximum parallelorum, æquales erunt anguli ſphærici C B A, D N K. </
s
>
<
s
xml:id
="
echoid-s3938
"
xml:space
="
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">Cum ergo an-
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gulo D N K, æqualis quoque ſit angulus D A K, per propoſ. </
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>
<
s
xml:id
="
echoid-s3939
"
xml:space
="
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">13. </
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>
<
s
xml:id
="
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"
xml:space
="
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">noſtrorum triangulorum ſphæricorum,
<
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quòd A D N, ABN, per propoſ. </
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>
<
s
xml:id
="
echoid-s3941
"
xml:space
="
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">11. </
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>
<
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xml:id
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xml:space
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">lib. </
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<
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xml:id
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xml:space
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">1. </
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<
s
xml:id
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xml:space
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">Theod. </
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>
<
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xml:space
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">ſemicirculi ſint; </
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>
<
s
xml:id
="
echoid-s3946
"
xml:space
="
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">æquales erunt anguli ſphærici F B A,
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F A B; </
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>
<
s
xml:id
="
echoid-s3947
"
xml:space
="
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">ac proinde & </
s
>
<
s
xml:id
="
echoid-s3948
"
xml:space
="
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">arcus B F, A F, æquales erunt, per propoſ. </
s
>
<
s
xml:id
="
echoid-s3949
"
xml:space
="
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">4. </
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>
<
s
xml:id
="
echoid-s3950
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xml:space
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">lib. </
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>
<
s
xml:id
="
echoid-s3951
"
xml:space
="
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">1. </
s
>
<
s
xml:id
="
echoid-s3952
"
xml:space
="
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">Menelai, vel per propoſ. </
s
>
<
s
xml:id
="
echoid-s3953
"
xml:space
="
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">40.
<
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/>
</
s
>
<
s
xml:id
="
echoid-s3954
"
xml:space
="
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">lib. </
s
>
<
s
xml:id
="
echoid-s3955
"
xml:space
="
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">3. </
s
>
<
s
xml:id
="
echoid-s3956
"
xml:space
="
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">Ioan. </
s
>
<
s
xml:id
="
echoid-s3957
"
xml:space
="
preserve
">Regiom. </
s
>
<
s
xml:id
="
echoid-s3958
"
xml:space
="
preserve
">de triangulis, vel certè per propoſ. </
s
>
<
s
xml:id
="
echoid-s3959
"
xml:space
="
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">9. </
s
>
<
s
xml:id
="
echoid-s3960
"
xml:space
="
preserve
">noſtrorum triangulorũ ſphæricorum. </
s
>
<
s
xml:id
="
echoid-s3961
"
xml:space
="
preserve
">Cum ergo
<
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/>
& </
s
>
<
s
xml:id
="
echoid-s3962
"
xml:space
="
preserve
">toti arcus B C, A D, æquales ſint, nempe quadrantes, vt oſtendimus; </
s
>
<
s
xml:id
="
echoid-s3963
"
xml:space
="
preserve
">erunt & </
s
>
<
s
xml:id
="
echoid-s3964
"
xml:space
="
preserve
">reliqui arcus F C, F D,
<
lb
/>
æquales. </
s
>
<
s
xml:id
="
echoid-s3965
"
xml:space
="
preserve
">Et quoniam ostenſi ſunt arcus E C, E D, æquales, erunt duo latera E C, E F, trianguli ſphærici
<
lb
/>
C E F, æqualia duobus lateribus E D, E F, trianguli ſphærici D E F. </
s
>
<
s
xml:id
="
echoid-s3966
"
xml:space
="
preserve
">Cum ergo habeant & </
s
>
<
s
xml:id
="
echoid-s3967
"
xml:space
="
preserve
">baſes C F,
<
lb
/>
D F, æquales, vt oſtenſum eſt, erunt per propoſ. </
s
>
<
s
xml:id
="
echoid-s3968
"
xml:space
="
preserve
">35. </
s
>
<
s
xml:id
="
echoid-s3969
"
xml:space
="
preserve
">lib. </
s
>
<
s
xml:id
="
echoid-s3970
"
xml:space
="
preserve
">3. </
s
>
<
s
xml:id
="
echoid-s3971
"
xml:space
="
preserve
">Ioan. </
s
>
<
s
xml:id
="
echoid-s3972
"
xml:space
="
preserve
">Regiom. </
s
>
<
s
xml:id
="
echoid-s3973
"
xml:space
="
preserve
">de triangulis, vel per propoſ. </
s
>
<
s
xml:id
="
echoid-s3974
"
xml:space
="
preserve
">
<
lb
/>
<
note
position
="
left
"
xlink:label
="
note-0080-08
"
xlink:href
="
note-0080-08a
"
xml:space
="
preserve
">40</
note
>
18. </
s
>
<
s
xml:id
="
echoid-s3975
"
xml:space
="
preserve
">noſtrorum triangulorum ſphæricorum, anguli C E F, D E F, æquales. </
s
>
<
s
xml:id
="
echoid-s3976
"
xml:space
="
preserve
">Diuidit ergo arcus E F, cir-
<
lb
/>
culi maximi angulum C E D, bifariam; </
s
>
<
s
xml:id
="
echoid-s3977
"
xml:space
="
preserve
">Ac propterea cum eundem bifariam ſecet arcus E G, vt de-
<
lb
/>
monſtratum eſt, tranſibit omnino arcus E G, productus per F, adeo vt ab arcu E F, non differat, ne
<
lb
/>
duo arcus concedantur eundem angulum C E D, bifariam ſecare. </
s
>
<
s
xml:id
="
echoid-s3978
"
xml:space
="
preserve
">Quapropter tres circuli horarij C F,
<
lb
/>
D F, E F, vnam eandem{q́ue} ſectionem habent communem. </
s
>
<
s
xml:id
="
echoid-s3979
"
xml:space
="
preserve
">Quod eſt propoſitum: </
s
>
<
s
xml:id
="
echoid-s3980
"
xml:space
="
preserve
">Acproinde in quo pun-
<
lb
/>
<
note
position
="
left
"
xlink:label
="
note-0080-09
"
xlink:href
="
note-0080-09a
"
xml:space
="
preserve
">Quænam horæ
<
lb
/>
ab or. vel occ.
<
lb
/>
& à mer. vel
<
lb
/>
med. noc. ſe mu
<
lb
/>
tuo ſecẽt in eo-
<
lb
/>
dem puncto.</
note
>
cto planum horologij communem hanc ſectionem ſecat, per idem communes ſectiones eorundem circulo-
<
lb
/>
rum, & </
s
>
<
s
xml:id
="
echoid-s3981
"
xml:space
="
preserve
">plani horologij tranſibunt, per propoſ. </
s
>
<
s
xml:id
="
echoid-s3982
"
xml:space
="
preserve
">18. </
s
>
<
s
xml:id
="
echoid-s3983
"
xml:space
="
preserve
">huius lib. </
s
>
<
s
xml:id
="
echoid-s3984
"
xml:space
="
preserve
">adeo vt in eodem puncto horologij ſe inter-
<
lb
/>
ſecent lineæ horariæ illorum circulorum. </
s
>
<
s
xml:id
="
echoid-s3985
"
xml:space
="
preserve
">Quocirca in quo puncto horaria linea circuli C F, horariam li-
<
lb
/>
neam circuli E F, ſecat, per idem ducenda erit linea horaria circuli D F, & </
s
>
<
s
xml:id
="
echoid-s3986
"
xml:space
="
preserve
">e contrario. </
s
>
<
s
xml:id
="
echoid-s3987
"
xml:space
="
preserve
">Quibus autem
<
lb
/>
horis deputentur circuli C F, D F, E F, docebit nos figura, quam in propoſ. </
s
>
<
s
xml:id
="
echoid-s3988
"
xml:space
="
preserve
">9. </
s
>
<
s
xml:id
="
echoid-s3989
"
xml:space
="
preserve
">huius libri poſuimus. </
s
>
<
s
xml:id
="
echoid-s3990
"
xml:space
="
preserve
">Si
<
lb
/>
<
note
position
="
left
"
xlink:label
="
note-0080-10
"
xlink:href
="
note-0080-10a
"
xml:space
="
preserve
">50</
note
>
enim alter circulorum C F, D F, nempe C F, tribuatur, verbi gratia, horæ 12. </
s
>
<
s
xml:id
="
echoid-s3991
"
xml:space
="
preserve
">ab ortu, vel occaſu; </
s
>
<
s
xml:id
="
echoid-s3992
"
xml:space
="
preserve
">& </
s
>
<
s
xml:id
="
echoid-s3993
"
xml:space
="
preserve
">
<
lb
/>
alter D F, exempli cauſa, horæ 20. </
s
>
<
s
xml:id
="
echoid-s3994
"
xml:space
="
preserve
">ab ortu, vel occaſu, erit E F, circulus horæ quartæ à meridie, vel me-
<
lb
/>
dia nocte, cum hæc hora in medio illarum ſit poſita in figura dicta propoſitionis 9. </
s
>
<
s
xml:id
="
echoid-s3995
"
xml:space
="
preserve
">huius libri, quemad-
<
lb
/>
modum & </
s
>
<
s
xml:id
="
echoid-s3996
"
xml:space
="
preserve
">punctum G, in medio punctorum C, & </
s
>
<
s
xml:id
="
echoid-s3997
"
xml:space
="
preserve
">D, poſitum eſt. </
s
>
<
s
xml:id
="
echoid-s3998
"
xml:space
="
preserve
">Sic etiam, ſi E F, ponatur eſſe
<
lb
/>
circulus horæ 1 {1/2}. </
s
>
<
s
xml:id
="
echoid-s3999
"
xml:space
="
preserve
">à meridie, vel media nocte, & </
s
>
<
s
xml:id
="
echoid-s4000
"
xml:space
="
preserve
">E C, horæ tertiæ ab ortu, vel occa-
<
lb
/>
ſu; </
s
>
<
s
xml:id
="
echoid-s4001
"
xml:space
="
preserve
">erit E D, circulus horæ 24. </
s
>
<
s
xml:id
="
echoid-s4002
"
xml:space
="
preserve
">ab ortu, vel occaſu, quòd hæ duę horę ab
<
lb
/>
illa hinc inde ęqualiter abſint, ſicuti & </
s
>
<
s
xml:id
="
echoid-s4003
"
xml:space
="
preserve
">puncta C, D, à puncto
<
lb
/>
G, ęqualibus interuallis diſiunguntur, &</
s
>
<
s
xml:id
="
echoid-s4004
"
xml:space
="
preserve
">c. </
s
>
<
s
xml:id
="
echoid-s4005
"
xml:space
="
preserve
">Ex his
<
lb
/>
<
note
position
="
left
"
xlink:label
="
note-0080-11
"
xlink:href
="
note-0080-11a
"
xml:space
="
preserve
">Compoſitio ſu-
<
lb
/>
periorum qua-
<
lb
/>
tuot, & ſequen-
<
lb
/>
tium duarum
<
lb
/>
& triginta ta-
<
lb
/>
bularum.</
note
>
autem nullo negotio conficiemus trigin-
<
lb
/>
ta ſex illas tabulas, quas in hoc
<
lb
/>
ſcholio conſcripſimus.</
s
>
<
s
xml:id
="
echoid-s4006
"
xml:space
="
preserve
"/>
</
p
>
</
div
>
</
text
>
</
echo
>