Clavius, Christoph, Geometria practica

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247217LIBER QVINTVS. cularis in pentagono inclinato cum prædicto latere Dodecaedri efficit. Ex quo
fit punctum P, in plano ſupremæ baſis exiſtere, atque idcirco perpendicularem
159[Figure 159] P Q, ad planum baſis per M N, ductum demiſſam, æqualem eſſe altitudini Do-
decaedri;
eiuſque ſemiſſem R Q, altitudini vnius pyramidis pentagonæ eſſe æ-
qualem.
Quæ omnia facil@ intelligentur, ſi Dodecaedrum aliquod materiale ad-
hibeatur.
Deniqve datum ſit Ico ſaedri latus a b, ſupra quod extruatur pentagonum
æquilaterum, &
æquiangulum a b c d e, pro baſe pyramidis ex quin que baſibus
Icoſaedri conflatæ.
Iuncta autem recta c e, ſeceturlatus a b, in ſ, bifariam, & re-
cta ducaturſd, quæ vt in Dodecaedro oſtendimus proximè, perpendicularis e-
rit ad vtramque a b, c e.
Fiat ſupra latus Icoſaedri c d, triangulum æquilaterum
c d h, probaſe vna Icoſaedri;
& diuiſo latere c d, bifariam in k, iungatur recta h-
k, quæ ad c d, erit perpendicularis.
Præterea ſupra c e, fiat Iſoſceles c g e, 11ſchol. 26.
Primi.
ius vtrum que laterum c g, e g, perpendicularihk, ſit æquale.
Poſt hæc ſupra ſ d,
conſtituatur triangulum ſdl, cuius latus ſl, perpendiculari h k, &
latus dl, lateri
Icoſaedri a b, fit æquale.
Denique angulo c g e, fiat æqualis angulus m n o, &
recta n o, perpendiculari h k, æqualis:
Item angulus n o p, angulo d l s, rectaque
o p, lateri Icoſaedri a b, æqualis.
Dico perpendicularem p q, ad m n, demiſſam,
eſſe altitudinem Icoſaedri, eiuſque ſemiſſem r q, altitudinem vnius pyramidis in
22Altitudo py-
ramidis Ico-
ſaedri.
Icoſaedro.
Quia enim, vt ex Hypſicle ad finem Euclidis demonſtrauimus, an-
gulus c g e, metitur in clinationem vnius baſis ad alteram, ſi m n, concipiatur eſ-
ſe perpendicularis, quæ in baſe infima Icoſaedri ex angulo trianguli ad medium
punctum lateris oppoſiti ducitur, reſpondebit n o, per pendiculari, quæ in trian-
gulo ad illam baſem inclinato ex eodem medio pũcto ad angulum oppoſitum
ducitur:
propterea quod angulum m n o, angulo inclinationis c g e, & rectam
n o, perpendiculari h k, æqualem fecimus:
Recta verò o p, referet latus Icoſae-
dri inter angulum dicti trianguli inclinati, &
angulum ſupremæ baſis poſitum;
propterea quod recta o p, poſita eſt æqualis lateri Icoſaedri, & angulus n o p,
angulo d l s:
qui quidem æqualis eſt illi, quẽ dictum latus efficit cum 338. primi. diculari ex angulo ſupradicti trianguli inclinati ad baſem, in medium punctum
lateris oppoſiti ducitur.
Eſt enim recta d s, æqualis perpendiculari ex angulo
pentagoni ad latus oppoſitum ductæ, &
latera sl, dl, æqualia

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