Full text: Cavalieri, Bonaventura: Geometria indivisibilibvs continvorvm

LIBER II. BM, angulus, HFE, æqualis eſt angulo illi coalterno, BCM, & ,
HEF, ipſi, FDC, qui eſt æqualis angulo illi oppoſito, FAC, qui
tandem æquatur angulo, MBC; interior exteriori, ideò anguius,
FEH, æquatur angulo, MBC, ſunt igitur in triangulis, FEH, M
BC, duo anguli duobus angulis æquales, & latera illis adiacentia
ſunt æqualia, nempè, FE, ipſi, BC, ergo reliqua latera erunt æ-
qualia, . ſ. HE, ipſi, BM, eodem modo oſtendemus de cæteris pa-
rallelis ipſi, CD, eas nempè, quæ verſus puncta, F, C, abſcindunt
à lateribus, FD, CA, partes æquales, eſſe pariter inter ſe æquales,
veiuti ſunt extremæ, AF, CD, æquales, ergo omnes lineæ trian-
guli, CAF, æquabuntur omnibus lineis trianguli, FDC, ſumptis
in vtriſq; omnibus lineis regula, CD, ergo triangulus, ACF, erit
æqualis triangulo, FDC, ergo duo trianguli, ACF, FDC, ſcili-
cet parallelogrammum, AD, erit duplum cuiuſuis triangulorum, A
CF, FCD, quod oſtendere opus erat.

219.1.

0166-02
26. Primi
Elem.
3. huius.

220. COROLLARIVM I.

_H_Inc patet, quæcunq; de parallelogrammis in Prop. 5. 6. 7. & 8. huius Librioſtenſaſunt, eadem de triangulis vt verarecipi poſſe,
ſi in triangulis conditiones ibi oppoſitæ repertæ fuerint, nam in vnoquo-
que expoſitorum triangulorum ſumptis duobus quibuſuis lateribus ſieri
poteſt ſub illis in eodem angulo parallelogr ammum, cuius triangulum
erit dimidium. Triangula ergo, quæ in eadem ſunt altitudine inter ſe
ſunt, vt baſes: Et quæ in eadem baſi mierſe ſunt, vt altitudines, vel vt
latera æqualiter baſibus inclinata; Item babent rationem compoſitam ex
ratione baſium, & altitudinum, ſine laterum æqualiter baſibus inclina-
torum, cum ſunt æquiangulæ: Item triangula, quorum baſes altitudi-
nibus, vel lateribus æqualiter baſibus inclinatis, reciprocantur ſunt æ-
qualia; & quæ ſunt æqualia baſes habent altitudinibus, vel lateribus
æqualiter baſibus inclmatis, reciprocas: Et tandem habetur ſimilia trian-
gula eſſe in dupla ratione laterum homologorum, quæ omnia ex præſenti
Propoſ. pendent.

220.1.

_Iux. diff._
_1. Sexti_
_Elem._

221. COROLLARIVM II.

_C_Olligitur in ſuper, ſi ſupponamur, CD, eſſe æqualem ipſi, DF,
quamlibet ductam in triangulo, FCD, parallelam ipſi, CD, æqua-
lem eſſe ei, quam ipſa abſcindit ab, FD, verſus, F, nempè ipſi abſciſſæ,
FE, & producta, EH, verjus, AC, cui incidat in, N, ipſam, HN,
æquari reſiduæ abſciſſæ, FE, . ſ. ipſi, ED, & , NE, integram æquari

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