Cavalieri, Buonaventura - Geometria indivisibilibvs continvorvm : noua quadam ratione promota

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7858GEOMETRIÆ GO, ſint verò in ambitu triangula, ABC, FIG; ABH, FIO, H
BC, OIG, ergo tria hęc tribus iam dictis ſimilia erunt, ergo & ba-
ſes, ACH, FGO, ſimiles erunt, nam cum ſit, AC, ad, CB, vt,
FG, ad, GI; BC, ad, CH, vt, IG, ad, GO, erit ex ęquali, AC,
ad, CH, vt, FG, ad, GO, eadem ratione oſtendemus, CH, ad,
115. Sexti
Elem. HA, eſſe vt, GO, ad, OF, ex quo habebitur ex ęquali, CA, ad,
AH, eſſe vt, GF, ad, FO, ergo triangula, ACH, FGO, ſimilia
erunt. Eodem modo probabimus triangula, AHD, FLO, ACD,
FGL, eſſe ſimilia, ex quo concludemus ipſas pyramides ſimiles eſſe.
Quod ſi tria triangula ad, B, I, terminantia omnia non ſint in am-
bitu, oſtendemus tamen illa eſſe ſimilia, erunt. n. vel baſes pyrami-
dum, quarum tria triangula verticalia erunt in ambitu, vel ſaltem
aliarum pyramidum, quarum triangula ſimilia eſſe probabuntur,
quia erunt baſes pyramidum tria triangula verticalia in ambitu ha-
bentium, ad hæc. n. tandem deuenire neceſſe erit: Igitur oſtenſum
eſt, quod proponebatur.

COROLLARIVM.

_Q_Via verò in pyramidibus triangulatis, BAHC, IFGO, exiſten-
tibus ſimilibus illarum triangulis verticalibus, baſes, ACH, F
GO, neceſſiriò ſimiles eſſe oſtenſæ ſunt, ideò ex hoc colligimus ſi
in duabus pyramidibus triangulatis tria verticalia triangulatribus ver-
ticalibus triangulis ſimilia ſint, etiam baſes ſimiles eſſe.

LEMMA V.

SI duo ſimilia triangula fuerint ſubiectis planis æquè ad eandem
partem inclinata, ita vt communes cum illis ſectiones ſint ea-
rum latera homologa, quæ tanquam baſes aſſumantur; ab eorum
autem verticibus rectæ lineæ in ſublimi fuerint conſtitutæ, angulos
æquales cum eorum lateribus homologis continentes, illæ erunt ſu-
biectis planis æqualiter inclinatæ, vel eiſdem ambo parallelæ; ſi au-
tem fuerint inclinatæ, & vſque ad ſubiecta plana producantur, iun-
ganturq; pucta occurſuum cum extremis baſium dictorum triangu-
lorum, pariter hinc conſtitutæ pyramides ſimiles erunt.

Sint ſimilia triangula, ABD, HPO, ſubiectis planis ęquè incli-
nata, in baſibus, BD, PO, à quorum verticibus, A, H, rectæ li-
neæ, AC, HN, in ſublimi conſtitutæ contineant cum homologis
eorum lateribus angulos æquales, ſint nempè anguli, CAB, NH
P, necnon, CAD, VHO, inter ſe æquales. Dico ipſas, AC, H
N, ſubiectis planis eſſe ęqualiter inclinatas, vel eiſdem ambo

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