1liter igitur vt ante oſtenderemus, vnamquamque qua
tuor pyramidum, quarum communis vertex S, baſes au
tem triangula, quæ ſunt circa pyramidem ABCD, eſse
quartam partem pyramidis ABCD. Siue igitur pun
ctum S, cadat intra vnam priorum quatuor pyrami
dum, ſiue in earum aliquo latere, ſeu triangulo; neceſ
ſario erit pars æquali toti; tam enim tota vna pyramis
quatuor priorum, quarum communis vertex F, quàm eius
pars, vna quatuor pyramidum poſteriorum, quarum com
munis vertex S, erit eiuſdem ABCD, pyramidis pars
quarta. Ex abſurdo igitur non in alio puncto à puncto F
ſecabunt ſe in eaſdem rationes quatuor rectæ, quæ ab angu
lis ad oppoſita triangula pyramidis ABCD, ducantur.
Manifeſtum eſt igitur propoſitum.
tuor pyramidum, quarum communis vertex S, baſes au
tem triangula, quæ ſunt circa pyramidem ABCD, eſse
quartam partem pyramidis ABCD. Siue igitur pun
ctum S, cadat intra vnam priorum quatuor pyrami
dum, ſiue in earum aliquo latere, ſeu triangulo; neceſ
ſario erit pars æquali toti; tam enim tota vna pyramis
quatuor priorum, quarum communis vertex F, quàm eius
pars, vna quatuor pyramidum poſteriorum, quarum com
munis vertex S, erit eiuſdem ABCD, pyramidis pars
quarta. Ex abſurdo igitur non in alio puncto à puncto F
ſecabunt ſe in eaſdem rationes quatuor rectæ, quæ ab angu
lis ad oppoſita triangula pyramidis ABCD, ducantur.
Manifeſtum eſt igitur propoſitum.
PROPOSITIO IX.
Omnis pyramis baſim habens triangulam di
uiditur in quatuor pyra mides æquales, & ſimiles
inter ſe, & toti, & vnum octaedrum totius pyrami
dis dimidium, ip ſi que concentricum.
uiditur in quatuor pyra mides æquales, & ſimiles
inter ſe, & toti, & vnum octaedrum totius pyrami
dis dimidium, ip ſi que concentricum.
Sit pyramis ABCD, cuius baſis triangulum ABC,
ſectisque omnibus lateribus bifariam, iungantur rectæ FG,
GH, HF, FK, KL, LM, MK, KH, HM, GL, LF.
Dico quatuor pyramides DKLM, LFBG, KHFA,
MHGC, æquales eſse, & ſimiles inter ſe, & toti pyrami
di ABCD: octaedrum autem eſse LFGMKH, & di
midium pyramidis ABCD, ipſique concentricum. Du
cantur enim rectæ DNH, BQH, LN: & poſita BE, du
pla ipſius BH, iungatur DOC, in triangulo DBH, &
ponatur DP, ipſius PE, tripla, & connectantur rectæ LP,
PH. Quoniam igitur E, eſt centrum trianguli ABC,
ſectisque omnibus lateribus bifariam, iungantur rectæ FG,
GH, HF, FK, KL, LM, MK, KH, HM, GL, LF.
Dico quatuor pyramides DKLM, LFBG, KHFA,
MHGC, æquales eſse, & ſimiles inter ſe, & toti pyrami
di ABCD: octaedrum autem eſse LFGMKH, & di
midium pyramidis ABCD, ipſique concentricum. Du
cantur enim rectæ DNH, BQH, LN: & poſita BE, du
pla ipſius BH, iungatur DOC, in triangulo DBH, &
ponatur DP, ipſius PE, tripla, & connectantur rectæ LP,
PH. Quoniam igitur E, eſt centrum trianguli ABC,
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