Full text: Cavalieri, Bonaventura: Geometria indivisibilibvs continvorvm

GEOMETRIÆ ſeuntis, vt ipſius, BDFO, quod ſemper eſt trapezium, & ipſarum, V
BO, TDF, ſiue eiſdem æquidiſtantium inter eaſdem ductarum, eſſe ea-
rundem lineas, vel latera homologa, vnde patet communes ſectiones
planiper latera fruſti conici ducti, & eiuſdem baſium oppoſitarum, ſiue
eiſdem æquidiſtantium inter eas productarum figurarum, eſſe earundem
lineas, vel latera homologa; lineas, inquam, cum ſunt intra figuras,
nec ſumuntur in plano tangente: latera, cum ſunt in earum circuitu,
cum nempè ſunt in eodem plano tangente, in eo præcisè, quod eſt pla-
num contactus fruſti conici (contactus ſcilicet cius plani, quod per ver-
ticem ducitur) quod ſemper erit trapezium, vel trapezia, vt patere po-
teſt in trapezijs, BDCR, IEFO, quæ eſſent planum contactus fruſti
conici, ſiidem fruſtum tangeretur à plano trianguli, ADF.

89. THEOREMA XIX. PROPOS. XXII.

SI duæ figuræ planę ſimiles, non exiſtentes in eodem pla-
no, fuerint inæquales, & ſimiliter poſitæ; erunt cuiu-
ſdam fruſticonici oppoſitæ baſes.

Vtamuradhuc figura Propoſ. 19. & ſint duæ figuræ planæ quæ-
cumque ſimiles, inæquales, & ſimiliter poſitę, non tamen exiſten-
tesin eodem plano, ipſæ, VBO, TDF. Dico, quod erunt am-
bæ cuiuſdam fruſti conici oppoſitę baſes. Quoniam ergo figure, V
BO, TDF, ſunt ſimiliter poſitæ, & non in eodem plano, erunt in
planis ęquidiſtantibus, & quia ſunt ſimiles ſint earum incldentes, & oppoſitarum tangentium, quæ ſunt earundem homologarum regu-
læ, ipſæ, KN, HP; KN, ipſius, VBO, & , HP, ipſius, TDF,
& prædictæ tangentes figuræ, VBO, ſint ipſæ, VK, XN, & fi-
guræ, TDF, ipſæ, TH, SP, erunt ergo ipſæ, KN, HP, æqui-
diſtantes, & quia ad tangentes, quæ ſunt regulæ homologarum, illę
efficiunt ad eandem partem angulos æquales, erit angulus, KNX,
æqualis angulo, HPS, & quia, KN, eſt parallela ipſi, HP, erit
etiam, XN, parallela ipſi, SP. Eodem pacto oſtendemus, VK,
eſſe parallelam ipſi, TH; ducantur in figuris, VBO, TDF, duæ
earum homologæ regulis dictis tang entibus, quæ ſint ipſæ, BR, I
O, DC, EF, ſint autem totæ, BO, DF, productæ, ſi opus ſit, vt
ſecent ipſas, KN, HP, quas diuident ſimiliter ad eandem partem,
vt in punctis, M, G, & quia figuræ propoſitæ ſunt inæquales, ſit
maior ipſa, TDF, igitur etiam maior erit, DC, ipſa, BR, vel, E
F, ipſa, IO, ſi, n. eſſent eiſdem æquales, etiam reliquæ homologæ
his parallelæ eſſent ęquales, cum omnes ſint proportionales (ſunt. n.

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