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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mutuo ſecent, ita palam fiet. </
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<
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xml:space
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">Si enim non ſe interſecent, cum ſint in eodem plano horologii, pa-
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rallelæ erunt. </
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<
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xml:space
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">Cum igitur vtraque H I, P Q, ipſi V X, parallela ſit, & </
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<
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P Q. </
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<
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<
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">H K, ipſi P Q, parallela, vt ſupra demonſtrauimus. </
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<
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xml:space
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ſe parallelæ erunt. </
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<
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xml:space
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">Quod eſt abſurdũ, cùm conueniant in H Secant ergo ſe mutuo P Q, V X. </
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<
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re V X, vtramque H λ, P Q, ſecans, per punctum λ, vbi ſe mutuo ſecant H λ, Q P, tranſibit. </
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<
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ſi verbi gratia, V X, ſecet rectam H λ, citra punctum λ, vel vltra, ſi producatur vlterius, ſtatim ſe-
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cabit planum circuli B C, in eodem puncto, in quo rectam H λ, ſecat, ac proinde ſecare non po-
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terit amplius rectam Q P, in eodem plano circuli B C, exiſtentem. </
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<
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xml:space
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">Pari ratione, ſi V X, ſecet P Q,
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citra, vel vltra punctum λ, ſecabit quoque producta ſtatim planum circuli B C, in eodem puncto,
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in quo rectam P Q, ſecat, ac propterea ſecare non poterit rectam H λ, in eodem ſemper plano
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circuli B C, exiſtentem. </
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<
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xml:space
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">Quare V X, niſi per punctum λ, tranſeat, vtramque H λ, Q P, non ſeca-
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bit, Quod eſt abſurdum. </
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<
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">oſtenſum eſt enim eam ſecare vtramque. </
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<
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">Eodem modo in δ, ſe mutuo
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ſecabunt tres rectæ α δ, θ ε, V X. </
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<
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xml:space
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">Quoniam igitur rectæ μ β, ξ γ, parallelæ ſunt, vtpote commu-
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nes ſectiones planorum parallelorum F G, M N, factæ à plano circuli D B C E, nec non & </
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<
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μ ξ, β γ, nempe communes ſectiones planorum parallelorum B C, Y Z, factæ à plano eodem
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xml:space
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circuli D B C E; </
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<
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">æquales erunt μ ξ, β γ. </
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<
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xml:space
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">Et quia H λ, μ ξ, necnon & </
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<
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">α δ, β γ, parallelæ etiam ſunt,
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propterea quòd anguli ω H λ, β α δ, ad puncta contactuum H, α, recti ſunt, (ſunt enim H μ, α β,
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diametri circulorum B C, Y Z, nempe communes ſectiones ipſorum, & </
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<
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polos bifariam ſecat, ex propo@. </
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<
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">anguli H μ ξ, α β γ, ad centra μ, β.
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</
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<
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">Cum enim per conſtructionem, recta B C, rectam H k, ad angulos rectos ſecet, rectus erit angu-
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lus H μ ξ. </
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<
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xml:space
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">Cum igitur rectis H μ, μ ξ, parallelæ ſint α β, β γ, erit & </
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<
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">angulus α β γ, ipſi H μ ξ, æqualis,
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rectus: </
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<
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">rectæ H μ, λ ξ, nec non α β, δ γ, parallelæ, nimirum ſectiones communes
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xml:space
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planorum parallelorum F G, M N, factæ à plano circuli B C, ſi de prioribus, & </
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<
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Y Z, ſi de poſterioribus loquamur; </
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<
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">fit vt recta δ ε, minor ſit, quàm recta λ P, vt mox demonſtra-
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bimus ſequenti lemmate. </
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<
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xml:space
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">Quare propinquior eſt recta V X, producta hyperbolæ O ε θ, in pun-
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cto δ, quàm in puncto P; </
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<
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xml:space
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">atque ita deinceps, ſi longius producantur linea horaria, & </
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le. </
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gens conicas ſuperficies in recta K L. </
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nocte plano horologii æquidiſtet, &</
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