Valerio, Luca - De centro gravitatis solidorvm libri tres

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PROPOSITIO XXXI.

Omnis pyramidis triangulam baſim habentis
idem eſt centrum grauitatis, & figuræ.

Sit pyramis ABCD, cuius baſis triangulum ABC,
centrum autem E. Dico E, eſſe centrum grauitatis pyra
midis ABCD. Secta enim ABCD, pyramide in quatuor
pyramides, ſimiles, & æquales inter ſe, & toti pyramidi
ABCD, & vnum octaedrum, ſint eæ pyramides DKLM,
MGCH, LBGF,
AKFH. Octaedrum
autem FGHKLM,
quod dimidium erit
pyramidis ABCD, &
ſint axes pyramidum
DSN, DS, KO, LP,
MQ: & ARG, iunga
tur. Quoniam igitur
FH, eſt parallela ipſi
BC, & ſecta eſt BC,
bifariam in puncto G,
tranſibit recta AG, per
centra triangulorum O,
& N, ad quæ axes KO,

45[Figure 45]
DN, terminantur; manifeſtum hoc eſt ex ſuperioribus:
eritque dupla AO, ipſius OR, nec non AN, dupla ipſius
NG, componendo igitur erit vt AG, ad GN, ita AR,
ad RO, & permutando, vt AG, ad AR, ita GN, ad
RO: ſed AG, eſt dupla ipſius AR, quoniam & AB, ip
ſius AF; igitur & GN, erit dupla ipſius RO: ſed & GN,
eſt dupla ipſius NR, nam N, eſt centrum trianguli GFH;
æqualis eſt igitur NR, ipſi RO, atque hinc dupla NO,

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