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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QI, quàm rectangulum ſub I R, R F, ad candem I F, applicatum, cxcedens{q́ue} quadrato
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ex R F, æquale eſſe quadrato ex A C, hoceſt, quartæ partirectanguli ſub F I, F O. </
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<
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xml:space
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ſcripto enim ex D I, quadrato D E, ducatur per Q, ipſi I E, parallela P N, occurrensrectæ
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G E, productæ in P, & </
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<
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xml:space
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">diametro G I, productæ in N, perficiatur{q́ue} figura, vt vides. </
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<
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xml:space
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niam igitur pallelogramma D E, M P, N I, circa eandem diametrum exiſtentia ſimilia
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<
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ſunt, eſt{q́ue} D E, quadratum; </
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xml:space
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0052-01
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que M P, N I, quadrata. </
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<
s
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xml:space
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">Et quoniam
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quadratum ex H L, æquale eſt quadratis
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ex H K, K L; </
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<
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xml:space
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">eſt autem recta H L, rectæ
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D Q, ſeu M N, & </
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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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">recta K L, rectæ D I, æqualis, ex con-
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ſtructione; </
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<
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xml:space
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">Erit quoque quadratũ M P,
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ex D Q, ſeu M N, deſoriptum, æquale
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quadrato D E, ex D I, deſcripto, vnà cũ
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quadrato ex A C. </
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<
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xml:space
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">Quare ablato commu-
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ni quadrato D E, erit@ reliqu{us} gnomon
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D N E, æqualis reliquo quadrato ex AC.
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</
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<
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Cum ergo gnomon D N E, æqualis ſit re-
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ctangulo F N, (Nam cum F M, ipſi M I,
<
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hoc eſt, ipſi I P, æquale ſit, addito communi M Q, æquale erit F N, gnomoni D N E,) eris
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<
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<
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quoque rectangulum F N, contcntum ſuh F Q, Q I, (quòdrecta Q I, rectæ Q N, æqua-
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lis ſit, ob quadratum I N, %%%% æquale quadrato ex A C, hoc eſt, quartæ parti quadrati ex AB,
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hoc eſt, rectanguli ſub F I, F O, comprehenſi. </
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>
<
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xml:space
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">Applicatum eſt ergo ad F I, diametrũ tranſ-
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uerſam rectangulum ſub F Q, Q I, æquale quartæ parti rectanguli ſub F I, F O, exce-
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dens quadrato rectæ Q I. </
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<
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xml:space
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">Eodem modo demonſtr abitur rectangulum ſub I R, R F, ap-
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plicatum ad F I, excedens{q́ue} quadrato ex R F, æquale eſſe quartæ parti rectanguli ſub F I,
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<
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F O. </
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</
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<
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xml:space
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">HIS præmiſſis, ſit F I, axis tranſuerſus duarum hyperbolarum oppoſitarum F G H, I K L, vt in
<
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">Alia deſcriptio
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hyperbolarum
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oppoſitarum@in
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plano.</
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figur a primi lemmatis, & </
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<
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">latus rectum F O, datum ex eodem primo lemmate, applicetur per ſecun-
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dum lemma ad F I, ex vtraque parte rectangulum ſub F Q, Q I, & </
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<
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">I R, R F, quartæ parti rectanguli
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ſub F I, F O, æquale, excedens{q́ue} quadrato ex I Q, & </
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<
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<
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">infra R, ſumantur vtcunque puncta
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quotlibet A, B, C, D. </
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<
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">Deinde ad interuallum I A, deſcribantur ex punctis Q, & </
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<
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">R, quatuor arcus,
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quos inpuncto E, ſecent alij quatuor arcus ex eiſdem punctis Q, & </
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<
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</
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<
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<
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">R, ad interuallum I B, quatuor arcus deſcribantur, quos in puncto G, in-
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terſecent alij quatuor ex eiſdem punctis Q, & </
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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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ualla I C, F C, ex punctis Q, & </
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<
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xml:space
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<
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ris punctis, ſi
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quaſint; </
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<
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xml:space
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">obſeruando ſemper, vt bini maiores arcus ex ſingulis quatuor, qui ex Q, & </
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<
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ſunt, deſcribantur ex Q, vltra punctum F, & </
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<
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citra punctum I, & </
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<
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<
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xml:space
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">Nam per puncta F, E, G, H, & </
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<
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">I, E, G, H, oppoſitæ
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hyperbolæ deſcribendæ erunt. </
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<
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xml:space
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">Quoniam enim recta Q E, hoc est, I A, ſuperat rectam E R, hoc eſt,
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F A, diametro tranſuerſa F I; </
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<
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xml:space
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<
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teris, tranſibunt hyperbolæ oppoſitæ, quarum axis F I, & </
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<
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xml:space
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doquidem, vt vult propoſitio 51. </
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<
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ctum Hyperboles inclinentur, maior minorcm ſuperat ipſo axe F I. </
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<
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xml:space
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& </
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<
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ſt, per K, ſecans rectam Q E, in K, ſiue in-
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<
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fra E, ſiue ſupra; </
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<
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<
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xml:space
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Q K, rectam K R, axe F I, per propoſ. </
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<
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<
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<
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ne recta Q E, rectam E R. </
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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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<
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<
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xml:space
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qui inter rectas K R, E R. </
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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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">, ſit recta E K, erit quoque eadem re-
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cta E K, exceſſus inter K R, & </
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<
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<
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">Quare recta E K, addita minori@earum, fiet aggregatum exhis
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duabus reliquæ æquale, ac proinde duo later a trianguli E K R, reliquo lateri æqualia erunt, ſed & </
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<
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<
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ra ſunt. </
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<
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xml:space
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">Quod eſt abſurdum. </
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<
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xml:space
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">Non ergo dicta Hyperbole per punctum K, ſed per E, tranſibit. </
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<
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pacto oſtendemus eandem pe
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r reliqua puncta G, H, &</
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<
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<
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</
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<
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<
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xml:space
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">MANIFESTVM autem eſt, deſcriptione
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m hanc ſolum conuenire conis rectis, vel etiam Sca-
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lenis, in quibus triangula per axem ad baſes conorum recta ſunt; </
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<
s
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