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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LIBER PRIMVS.
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fiet, & </
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<
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<
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xml:space
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<
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xml:space
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<
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xml:space
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">Igitur vt antea, iterum notus erit angu-
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lus P R T, altitudinis poli, & </
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<
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xml:space
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">PROBLEMA 8. PROPOSITIO 29.</
head
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<
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<
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<
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xml:space
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">Horizontem, vel ad Meri-
<
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/>
dianum tantum, vel ad Horizontem tantum inclinato, quanta ſit poli
<
lb
/>
altitudo ſupra ipſum, deprehendere.</
s
>
<
s
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="
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xml:space
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"/>
</
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<
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">10</
note
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<
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<
s
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xml:space
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">SIT planum circuli A B C D, cuius centrum E, & </
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<
s
xml:id
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xml:space
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">ad Meridianum, & </
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>
<
s
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xml:space
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">ad Horizontem, vel ad
<
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/>
<
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="
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xlink:label
="
note-0119-02
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xlink:href
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note-0119-02a
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xml:space
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">Altitudo poli
<
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ſupra planum
<
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/>
inclinatum ad
<
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/>
Meridianum,
<
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& Horizontem,
<
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vel ad Meridia-
<
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num t
<
unsure
/>
antum,
<
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/>
quomodo inue
<
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/>
miatur.</
note
>
Meridianum tantum inclinatum, & </
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<
s
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xml:space
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">communis ipſius, ac Meridiani ſectio B D. </
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>
<
s
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xml:space
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">Inuento autem,
<
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ex coroll. </
s
>
<
s
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="
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xml:space
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">propoſitionis præcedentis, arcu Meridiani inter planum inclinatum, & </
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>
<
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xml:space
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">polum mun-
<
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<
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fig-0119-01
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="
fig-0119-01a
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number
="
84
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<
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file
="
0119-01
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xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0119-01
"/>
</
figure
>
di arcticum, ſumatur illi ęqualis D F. </
s
>
<
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xml:space
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">Inueniatur
<
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quoque per coroll. </
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>
<
s
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xml:space
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">propoſ. </
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>
<
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xml:space
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">25. </
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>
<
s
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xml:space
="
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">huius lib. </
s
>
<
s
xml:id
="
echoid-s5667
"
xml:space
="
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">minor
<
lb
/>
diameter Ellipſis, quam perpendiculares ex cir-
<
lb
/>
cunferentia Meridiani in planum inclinatum de-
<
lb
/>
miſſæ faciunt, quæ ſit G H, ſecans maiorem B D,
<
lb
/>
ad angulos rectos, & </
s
>
<
s
xml:id
="
echoid-s5668
"
xml:space
="
preserve
">circa G H, circulus deſcri-
<
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batur, cuius circunferentiam ſecet recta ducta E F,
<
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/>
<
note
position
="
left
"
xlink:label
="
note-0119-03
"
xlink:href
="
note-0119-03a
"
xml:space
="
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">20</
note
>
in I. </
s
>
<
s
xml:id
="
echoid-s5669
"
xml:space
="
preserve
">Deinde per F, agatur minori diametro paral-
<
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/>
lela F K; </
s
>
<
s
xml:id
="
echoid-s5670
"
xml:space
="
preserve
">per I, autem maiori diametro parallela
<
lb
/>
I K, ſecans priorem in K, puncto, per quod dia-
<
lb
/>
meter ducatur A C, ad quam ex K, perpendicula-
<
lb
/>
ris erigatur K L, ſecans circulum A B C D, in L.
<
lb
/>
</
s
>
<
s
xml:id
="
echoid-s5671
"
xml:space
="
preserve
">Dico arcum C L, æqualem eſſe arcui, qui altitudi-
<
lb
/>
nem poli ſupra planum A B C D, metitur. </
s
>
<
s
xml:id
="
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xml:space
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">Quo-
<
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niam enim arcus D F, ęqualis eſt ar cui Meridiani
<
lb
/>
inter planum A B C D, & </
s
>
<
s
xml:id
="
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xml:space
="
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">polum mundi, erit & </
s
>
<
s
xml:id
="
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"
xml:space
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">
<
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/>
reliquus F O, reliquo in Meridiano à polo vſque
<
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/>
<
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position
="
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="
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note-0119-04a
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">30</
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>
ad diametrum, quæ ipſam B D, ſecat ad angulos rectos, & </
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>
<
s
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="
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xml:space
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">à qua perpendiculares cadunt in puncta
<
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G, H, æqualis. </
s
>
<
s
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xml:space
="
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">Quare per ea, quæ propoſ. </
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>
<
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xml:space
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">26. </
s
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<
s
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xml:space
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">huius lib. </
s
>
<
s
xml:id
="
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xml:space
="
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">demonſtrata ſunt, cadet perpendicularis
<
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/>
ex polo in planum A B C D, demiſſa in punctum K, ellipſis diametrorum B D, G H. </
s
>
<
s
xml:id
="
echoid-s5680
"
xml:space
="
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">Sicut enim in
<
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/>
figura illius propoſitionis ſe habent arcus D L, L E, quibus in circulo inclinato ęquales ſunt ar-
<
lb
/>
cus D K, K A, ita hic ſe habent arcus D F, F O, quibus in Meridiano ad circulum A B C D, incli-
<
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/>
nato reſpondent arcus ęquales à D, vſque ad polum arcticum, & </
s
>
<
s
xml:id
="
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xml:space
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">à polo vſque ad diametrum, quę
<
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ipſam B D, ad angulos rectos ſecat. </
s
>
<
s
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="
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xml:space
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">Quare vt ibi demonſtratum eſt, perpendicularem ex K, demiſ-
<
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ſam cadere in punctum Q, vbi ſe interſecant rectæ L Q, M Q, diametris H I, B D, ellipſis æqui
<
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/>
diſtantes, ita quoque hic oſtendetur, perpendicularem ex polo demiſſam cadere in punctum K,
<
lb
/>
vbi ſe interſecant rectæ F K, I K, diametris G H, B D, ellipſis æquidiſtantes. </
s
>
<
s
xml:id
="
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xml:space
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">Sit igitur perpendi-
<
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<
note
position
="
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="
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="
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xml:space
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">40</
note
>
cularis à polo cadens K M, & </
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<
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xml:space
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">polus M; </
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>
<
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xml:space
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">intelligaturq́; </
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>
<
s
xml:id
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xml:space
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">circulus maximus A M C, duci per rectas
<
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A E K C, K M, qui neceſſario ad planum A B C D, rectus erit; </
s
>
<
s
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="
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xml:space
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">ac propterea cum per polum mun
<
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<
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position
="
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xlink:label
="
note-0119-06
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xlink:href
="
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xml:space
="
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">18. vndec.</
note
>
di M, tranſeat, inſtar Meridiani erit ipſius plani inclinati, recta autẽ A C, linea erit meridiana, & </
s
>
<
s
xml:id
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<
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arcus C M, altitudinẽ poli ſupra idem planum metietur. </
s
>
<
s
xml:id
="
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xml:space
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">Ducantur quoque rectæ E L, E M, C L,
<
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C M. </
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>
<
s
xml:id
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xml:space
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">Quoniam igitur tam quadratum ex E L, quadratis ex E K, K L, quam quadratum ex E M,
<
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/>
<
note
position
="
right
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xlink:label
="
note-0119-07
"
xlink:href
="
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xml:space
="
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">47. primi.</
note
>
quadratis ex E K, K M, æquale eſt; </
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<
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xml:space
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">propterea quòd anguli E K L, E KM, recti ſunt, ex conſtru-
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ctione, & </
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>
<
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xml:space
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<
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<
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<
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<
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">Euclidis: </
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<
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xml:space
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">Sunt autem quadrata rectarum E L, E M, æqualium ex cen-
<
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tro ſphæræad eius ſuperficiem ductarum ęqualia; </
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>
<
s
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xml:space
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">erunt & </
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>
<
s
xml:id
="
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xml:space
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">quadrata ex E K, K L, quadratis ex E K,
<
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K M, ęqualia. </
s
>
<
s
xml:id
="
echoid-s5700
"
xml:space
="
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">Dempto ergo communi quadrato ex E K, æquale erit quadratum ex K L, quadrato
<
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/>
ex K M, & </
s
>
<
s
xml:id
="
echoid-s5701
"
xml:space
="
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">recta K L, rectæ K M, æqualis. </
s
>
<
s
xml:id
="
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xml:space
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">Itaque cum latera K L, K C, lateribus K M, K C, ſint
<
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<
note
position
="
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xlink:label
="
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xlink:href
="
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xml:space
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">50</
note
>
æqualia, angulosq́; </
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<
s
xml:id
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xml:space
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">æquales cõprehendant, vtpote rectos, æqualis erit baſis C L, baſi C M; </
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>
<
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xml:space
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">ac proin
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<
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>
de & </
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<
s
xml:id
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xml:space
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">arcus C L, æqualis erit arcui C M, qui altitudinẽ poli ſupra planũ A B C D, metitur. </
s
>
<
s
xml:id
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xml:space
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<
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<
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>
eſt propoſitum: </
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<
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xml:space
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<
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de plano ad Horizontem tantum inclinato nihil dicat. </
s
>
<
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xml:space
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">Quod idẽ nos ex ſinubus ita abſoluemus.</
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<
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</
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<
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xml:space
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">Altitudo poli
<
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ſupra planum
<
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inclinatum ad
<
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/>
Meridianum, &
<
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/>
Horizon@ẽ qua
<
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/>
uia per ſinu@ in
<
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quiratur.</
note
>
<
p
>
<
s
xml:id
="
echoid-s5710
"
xml:space
="
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">SIT Horizon A B C D, Meridianus A C; </
s
>
<
s
xml:id
="
echoid-s5711
"
xml:space
="
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">planum ad Meridianum & </
s
>
<
s
xml:id
="
echoid-s5712
"
xml:space
="
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">ad Horizontem inclina
<
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tum E F, ſec
<
unsure
/>
ans Meridianum in G, vbicunque hoc contingat; </
s
>
<
s
xml:id
="
echoid-s5713
"
xml:space
="
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">Polus mundi H, per quem & </
s
>
<
s
xml:id
="
echoid-s5714
"
xml:space
="
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">polũ
<
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/>
plani inclinati E F, circulus maximus deſcribatur B D, ſecans planum inclinatum in I, atque adeo
<
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per propoſ. </
s
>
<
s
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="
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">15. </
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>
<
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">lib. </
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>
<
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">1. </
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>
<
s
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="
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"
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">Theodoſij, ad angulos rectos; </
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>
<
s
xml:id
="
echoid-s5719
"
xml:space
="
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">metieturq́; </
s
>
<
s
xml:id
="
echoid-s5720
"
xml:space
="
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">propterea arcus HI, altitudinem
<
lb
/>
poli ſupra planum E F. </
s
>
<
s
xml:id
="
echoid-s5721
"
xml:space
="
preserve
">Quoniam igitur in triangulo ſphęrico G H I, cuius angulus I, rectus eſt,
<
lb
/>
vt ſinus arcus Meridiani G H, qui inter planum inclinatum, & </
s
>
<
s
xml:id
="
echoid-s5722
"
xml:space
="
preserve
">polum interijcitur, ad ſinum angu
<
lb
/>
li recti I, hoc eſt, ad ſinum totum, ita eſt, per propoſ. </
s
>
<
s
xml:id
="
echoid-s5723
"
xml:space
="
preserve
">16. </
s
>
<
s
xml:id
="
echoid-s5724
"
xml:space
="
preserve
">lib. </
s
>
<
s
xml:id
="
echoid-s5725
"
xml:space
="
preserve
">4. </
s
>
<
s
xml:id
="
echoid-s5726
"
xml:space
="
preserve
">Ioan. </
s
>
<
s
xml:id
="
echoid-s5727
"
xml:space
="
preserve
">Regiom. </
s
>
<
s
xml:id
="
echoid-s5728
"
xml:space
="
preserve
">de triangulis, vel </
s
>
</
p
>
</
div
>
</
text
>
</
echo
>