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
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gulum B C, ſub lateribus trianguli per axem comprehenſum; </
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<
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xml:space
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ad rectangulum ſub lateribus A B, A C, contentum, ita E K, ad A E. </
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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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">Apollonij, E K, latus rectum eſt paraboles E F G, hoc eſt, Recta, iuxta quam poſ-
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ſunt ordinatim applicatæ, & </
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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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">INVENTO igitur latere recto, ſumatur in plano aliquo axis parabolæ quicunque E H. </
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<
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<
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xml:space
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Paraboles in
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plano.</
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illa enim Parabola hic agimus, cuius diameter etiam axis eſt, ſecans omnes ordinatim applicatas bifa-
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riam, & </
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<
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">ad rectos angulos) in quo ſumantur quot cunque partes inter ſe æquales, (quò autem minores hæ
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partes fuerint, eò accuratius parabola deſcribetur) ita vt E A, ſit 1; </
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<
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<
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<
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deinceps, ſecundum numerorum imparium ſeriem: </
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<
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xml:space
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">atque per puncta A, B, C, H, &</
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<
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<
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xml:space
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">ad E H, perpen-
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diculares vtrinque ducantur eo modo, quo ſupra docuimus. </
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<
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xml:space
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">Deinde inter latus rectum E k, & </
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<
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E A, inuenta media proportionali, abſcindatur ei vtrinq; </
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<
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xml:space
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">æqualis A D; </
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<
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xml:space
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">& </
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<
s
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xml:space
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">ex B, vtrinq; </
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<
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B F, dupla ipſius A D; </
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<
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xml:space
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">& </
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>
<
s
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xml:space
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">ex C, vtrinque C G, tripla eiuſdem A D, & </
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<
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xml:space
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">ex H, ipſa H I, quadrupla, & </
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<
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ſic deinceps ſecundum naturalem ſeriem numerorum. </
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<
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xml:space
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">Nam per puncta D, F, G, I, deſcribenda erit para-
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bola. </
s
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<
s
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xml:space
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">Quod enim per punctum D, tranſeat, ex eo probatur, quod quadratum ex A D, recta, quæ media
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proportionalis est inter E K, E A, æquale eſt rectangulo ſub E K, E A, atque adeò A D, ordinatim
<
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<
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applicata eſt in parabola, cuius latus rectum E K, vt conſtat ex propoſ. </
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<
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<
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<
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">Apollonij. </
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<
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rabola per punctum D, tranſibit. </
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<
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">Si enim per aliud punctum, vt per P, tranſiret, eſſet quadratum quc
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-
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que ex A P, rectangulo ſub E K, E A, æquale, ex propoſ. </
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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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plicata eſſet ad diametrum. </
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<
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">Quare quadrata ex A D, A P, æqualia, & </
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<
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& </
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<
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">totum, quod eſt abſurdum. </
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<
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<
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autem tranſeat quoque per puncta F, G, I, ita oſtendemus. </
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<
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">Quoniam recta B F, dupla eſt rectæ A D,
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habebit quadr at um illius ad huius quadr at um proportionem quadruplam; </
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<
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xml:space
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<
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plicatam proportionem laterum) quemadmodum & </
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<
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<
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<
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recta C G, rectæ A D, tripla est, erit quadratum illius noncuplum quadrati huius, ſicut & </
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<
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noncupla eſt rectæ E A. </
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<
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xml:space
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">Eodem modo habebit quadratum ex H I, ad quadratum ex A D, eandem pro-
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portionem, quam recta E H, ad E A, nempe ſedecuplam, & </
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<
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</
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<
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<
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<
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">Apollonij, parabola per puncta F, G, I, tranſibit. </
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<
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xml:space
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">Nam ſi per aliud punctum, vt per Q, tranſi-
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re dicatur, erit ex dicta propoſ. </
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<
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<
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<
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<
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">Apollonij, quadratum ex B Q, ad quadratum ex A D, vt recta
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E B, ad rectam E A, hoc eſt, vtquadratum ex B F, ad quadratum ex A D. </
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<
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B Q, & </
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<
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<
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<
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<
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ſed per F, deſcribenda erit, & </
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<
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</
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<
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<
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<
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metro E H, inter C, H, puncta, punctum aliquod, quod terminet particulas diametri, quas quaterna-
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rius numeret, vt 8, vel 12. </
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<
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<
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">&</
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<
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<
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">cuiuſmodiest punctum M, terminans duodecim parti-
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culas. </
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<
s
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xml:space
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">Deinde lineæ E M, ſumemus quartam partem, vt in dato exemplo rectam E L, continentem
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tres particulas, & </
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<
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xml:space
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">ex L, perpendicularem ducemus ad E H, nempe L N, quæ parabolam ſecet in N, pun
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cto. </
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<
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xml:space
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">Si enim per M, ducamus aliam perpendicularem ad E H, ex qua abſcindamus M O, duplam ipſius
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L N, tranſibit parabola per punctum O; </
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<
s
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xml:space
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">propterca quod L M, ipſius E L, tripla eſt, & </
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<
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xml:space
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L N, dupla, quemadmodum & </
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<
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<
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<
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</
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<
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<
s
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">HAEC eadem ratio accommodari poteſt Parabolæ, in qua ordinatim applicatæ non ſunt perpendi-
<
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culares ad diametrum E H, vt in conis ſcalenis contingit, cum triangulum per axem ad baſim conirectũ
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non eſt, vt ex 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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<
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xml:space
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<
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de per puncta A, B, C, H, ducendæ erunt lineæ inter ſe parallelæ, facientes cum diametro E H, </
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