Full text: 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

GNOMONICES erit B M, circunferentia horaria. Si denique ex puncto n, vbi diameter paralleli diametrum Ver-
ticalis ſecat, vt centro, interuallo verò n K, in Meridiano ſumatur beneficio circini punctum P,
erit A P, circunferentia
deſcenſiua. Ratio hu-
ius rei eſt, quòd ducta
recta f L, perpendicula
ris eſt ad rectam ELY: ducta autẽ recta MLN,
ad B D, perpẽdicularis
eſt; & recta P L O, ad
A C, vt mox demon-
ſtrabimus. Cum ergo
prius per has perpendi
culares L f, M L N,
P L O, inuentę ſint tres
dictæ circunferentię, vt
in ſequenti cap. oſten-
demus, eædem etiam
inuentæ erunt per pun-
cta f, M, P, in Meridia
no accepta, vt diximus. Rectam autem f L, ad
E L Y, perpendicularẽ
eſſe, ita probabimus. Ducta recta E f, quoniã
duo latera K L, L E, triã
guli K L E, ęqualia ſunt
duobus lateribus f L,
L E, trianguli f L E, (quòd interuallum L f, interuallo L K, ſumptum eſt æquale) eſtq́ue baſis k E,
baſi f E, ęqualis, (quòd vtraque ſit ſphęrę ſemidiameter) erit angulus k L E, angulo f L E, ęqualis. Cum ergo k L E, rectus ſit, vt
paulo ante oſtendimus, erit & f L E, rectus, ideoq́; fL. , ad ELY,
perpendicularis erit. Viciſſim
etiam probabimus, ſi ex L, duca
tur ad E L Y, perpendicularis
L f, eam æqualẽ eſſe rectæ L K. Cũ enim duo quadrata ex E k,
E f, æqualia ſint, erunt duo qua-
drata ex E L, L K, duobus qua-
dratis ex E L, L f, æqualia Abla
to ergo communi quadrato re-
ctæ E L, reliqua erunt quadra-
ta rectarum L K, L f, æqualia,
proptereaq́ue & rectæ L K, L f,
æquales erunt. Quod etiam ita
confirmabimus. Extendatur re-
cta Y E, vſque ad Z. Quoniam
igitur K L, ad diametrum paral
leli a b, perpendicularis media
propottionalis eſt inter ſegmen
ta a L, Lb, ex ſcholio propoſ. 13. lib. 6. Eucl. erit quadratum ex
K L, æquale rectangulo ſub a L,
Lb, contento. Eodem modo, erit fL, perpendicularis ducta ad Y Z, media proportionalis inter
ſegmenta Y L, L Z, atque adeo quadratum ex fL, rectangulo ſub γ L, L Z, æquale. Cum ergo re-
ctangula ſuba L, L b, & ſub γ L, L Z, æqualia ſint, erunt & quadrata ex K L, f L, æqualia, ideoq́ue
& rectæ K L, f L, æquales.

472.1.

Alia inuentio
circunferentiæ
hectemotiæ, ho-
rariæ, & deſcen-
ſiuæ.
0554-01
10
20
8. primi.
30
0554-02
47. primi.
40
50
17. ſexti.
35. tertij.

AT vero rectas M L N, P L O, ad rectas B D, A C, perpendiculares eſſe, facile comprobabi-
mus, ſi prius demonſtremus, ſi per L, ducantur rectæ M L N, P L O, ad B D, A C, perpendicula-
res, coniungaturq́ue rectæ d M, d K, & n P, n K, rectam d M, rectæ d K, & rectam n P, rectæ

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