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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culorum A C, B D, G H. </
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>
<
s
xml:id
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xml:space
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">Deſcribatur enim maximus parallelorum L M, qui per puncta E, F, tran
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ſibit, cum per præcedentem propoſ. </
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>
<
s
xml:id
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xml:space
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">E F, ſit communis ſectio trium circulorum A C, B D, L M,
<
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propterea quòd A C, BD, tangunt parallelum B C,
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<
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fig-0071-01
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number
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53
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<
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xlink:href
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0071-01
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</
figure
>
in punctis oppoſitis, & </
s
>
<
s
xml:id
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echoid-s3360
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xml:space
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">L M, eſt parallelorum ma-
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ximus. </
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<
s
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xml:space
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">Deſcripto autem per polos parallelorum,
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& </
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>
<
s
xml:id
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xml:space
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preserve
">per contactum B, circulo maximo G L H M,
<
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tranſibit hic idem per polos quoque circuli B D,
<
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per propoſ. </
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<
s
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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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">2. </
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<
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xml:space
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">Theodoſii. </
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<
s
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xml:space
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">Quare ſecabit
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ſegmenta B E F, L E F, per propoſ. </
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<
s
xml:id
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xml:space
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">9. </
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<
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xml:space
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">eiuſdem,
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bifariam. </
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<
s
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xml:space
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">Cum ergo L E F, ſemicirculus ſit, quòd
<
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<
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xml:space
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">10</
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maximi circuli ſe bifariam ſecent, per propoſ. </
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<
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xml:space
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</
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<
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xml:space
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<
s
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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, erunt arcus L E, L F, quadran-
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res. </
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<
s
xml:id
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"
xml:space
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">Quoniam verò circuli maximi G L H M,
<
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G E H F, per G, H, polos parallelorum B C,
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L M, deſcripti ſunt, erunt per propoſ. </
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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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<
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Theodoſii, arcus inter ipſos intercepti ſimiles: </
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>
<
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xml:space
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autem arcus B I, B k, paralleli B C, inter ipſos inter-
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cepti, ex hypotheſi, quadrantes. </
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>
<
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xml:space
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">Igitur & </
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>
<
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ralleli L M, intercepti inter eoſdem, quadrantes e-
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runt: </
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>
<
s
xml:id
="
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xml:space
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">ac proinde, cum L E, L F, oſtenſi ſint quadrantes, tranſibit circulus G H, per puncta E, F,
<
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<
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atque adeo vtrunque circulum A C, B D, per rectam E F, ſecabit. </
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<
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xml:space
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">Quare recta E F, communis
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eſt ſectio trium circulorum A C, B D, G H. </
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<
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xml:space
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">Quod oſtendendum erat.</
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>
<
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</
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<
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<
s
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xml:space
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">TANGANT deinde in eadem Sphæra A B C D, eundem parallelum B C, in punctis non
<
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oppoſitis E, F, duo circuli maximi E G, F H, quorum communis ſectio ſit recta I K, quæ diameter
<
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erit ipſorum, cum per propoſ. </
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<
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<
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<
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<
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<
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file
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0071-02
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xlink:href
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0071-02
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</
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>
mutuo bifariam ſecent: </
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>
<
s
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xml:space
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">Secet autem eundem pa-
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rallelum B C, alius circulus maximus L M, per pa-
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ralleli polos L, M, & </
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>
<
s
xml:id
="
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"
xml:space
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">per axem L M, incedens, in
<
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punctis O, P, æqualiter diſtantibus à punctis E, F,
<
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ita vt arcus O E, O F, & </
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>
<
s
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xml:space
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">P B E, P C F, æquales ſint.
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</
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<
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<
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position
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Dico circulum L M, ſecare vtrumque maximum
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E G, F H, per rectam I k, hoc eſt, tranſire per
<
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puncta I, K, ita vt recta I K, ſit communis ſectio
<
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trium circulorum maximorum E G, F H, L M.
<
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</
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<
s
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xml:space
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">Cum enim circulus L M, ſecet parallelum B C,
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per polos, ſecabit ipſum, per propoſ. </
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<
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<
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<
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<
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Theodoſii, bifariam. </
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<
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xml:space
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">Sectio igitur communis O P,
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diameter erit paralleli B C, tranſiens per centrum
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Q, in quod axis L M, cadit, per propoſ. </
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<
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<
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xml:space
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<
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<
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Theodoſii. </
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<
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xml:space
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">Sit quoque R, centrum ſphęrę, per quod
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<
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& </
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<
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">axis L M, & </
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<
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xml:id
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xml:space
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">I K, diameter circulorum maximorũ
<
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tranſit. </
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<
s
xml:id
="
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xml:space
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">Et quia circuli in Sphęra ſe mutuo tangere dicuntur, cum communis ſectio planorum, in
<
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quibus circuli exiſtunt, vtrum que circulum tangit, ex defin. </
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>
<
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xml:id
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">lib. </
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<
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<
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xml:space
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">Theodofii: </
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>
<
s
xml:id
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xml:space
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">Sint communes ſe-
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ctiones circulorum E G, B C, & </
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>
<
s
xml:id
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xml:space
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">F H, B C, rectę E S, F S, tangentes ipſos circulos. </
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>
<
s
xml:id
="
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xml:space
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">Ductis ergo è
<
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centro Q, ſemidiametris Q E, Q F, erunt anguli S E Q, S F Q, recti, & </
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<
s
xml:id
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xml:space
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">idcirco, ducta recta E F,
<
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<
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xlink:label
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xml:space
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">18. tertij.</
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>
anguli S E F, S F E, rectis minores. </
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<
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xml:space
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">Quare rectæ E S, F S, in eodem plano paralleli B C, quem tan
<
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gunt, exiſtentes conuenient in aliquo puncto, vtpote in S, per 11. </
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<
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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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">Et quoniam K I, communis ſectio circulorum E G, F H, conuenit quoque cum vtraque E S, F S,
<
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vt mox oſtendemus lemmate ſequenti; </
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>
<
s
xml:id
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xml:space
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">fit vt KI, producta vtrique occurrat in S. </
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<
s
xml:id
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xml:space
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">Nam ſi alteram ip
<
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ſarũ ſecaret infra, aut ſupra S, non coiret cũ reliqua, vt patet. </
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>
<
s
xml:id
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xml:space
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">Si enim K I, occurrat, verbi gratia, re-
<
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<
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ctæ E S, alibi, quàm in puncto S, ſecabit ea producta ſtatim planũ circuli B C, in eo puncto, in quo
<
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rectam E S, ſecat, ac proinde nullo modo ſecabit rectam F S, in plano eodem circuli BC, exiſtentẽ.
<
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</
s
>
<
s
xml:id
="
echoid-s3425
"
xml:space
="
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">Et ſi K I, occurrat rectæ F S, alibi quàm in puncto S, oſtendemus eodẽ modo, ipſam ſecare nõ poſ-
<
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ſe rectam E S. </
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>
<
s
xml:id
="
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xml:space
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">Quamobrem recta K I, niſi per punctum S, tranſeat, non ſecabit vtramque E S, F S,
<
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Quod eſt abſurdum. </
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>
<
s
xml:id
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xml:space
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">Vtramque enim ſecat, vt in lemmate ſequenti oſtendemus. </
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>
<
s
xml:id
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xml:space
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">Ducantur iam
<
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rectæ E O, F O, O S, in plano circuli B C. </
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>
<
s
xml:id
="
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xml:space
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">Quia igitur arcus E O, F O, ponuntur æquales, æqua-
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les erunt & </
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>
<
s
xml:id
="
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xml:space
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">rectæ E O, F O: </
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>
<
s
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xml:space
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">Sunt autem & </
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>
<
s
xml:id
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">tangentes S E, S F, per 2. </
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<
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<
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<
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<
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">lib. </
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<
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xml:space
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">3. </
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>
<
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xml:space
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">Eu-
<
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<
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">28. tertij.</
note
>
clidis, æquales. </
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>
<
s
xml:id
="
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xml:space
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">Igitur erunt duo latera E O, O S, trianguli E O S, duobus lateribus F O, O S, trian
<
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guli F O S, æqualia, & </
s
>
<
s
xml:id
="
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xml:space
="
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">baſis E S, baſi F S; </
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>
<
s
xml:id
="
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xml:space
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">ac proinde & </
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>
<
s
xml:id
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xml:space
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">anguli E O S, F O S, æquales erunt. </
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>
<
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xml:space
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">Non
<
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<
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xlink:label
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">8. primi.</
note
>
aliter oſtendemus angulos E O Q, F O Q, æquales eſſe; </
s
>
<
s
xml:id
="
echoid-s3444
"
xml:space
="
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">propterea quòd latera E O, O Q, trian-
<
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guli E O Q, lateribus F O, O Q, trianguli F O Q, æqualia ſunt, & </
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>
<
s
xml:id
="
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="
preserve
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