Full text: Cavalieri, Bonaventura: Geometria indivisibilibvs continvorvm

LIBER VI. AG, & per, β, ducatur, βΔ, parallela ipſi, QP, ſecans curuam pa-
rabolæ in, Ω, & iunctis, LΩ, producatur, LΩ, vſq; ad, QP, cui in-
cidat in, Σ. Quia igitur eſt, QP, ad, ΡΣ, vt, PL, ad, Lβ, per con-
uerſionem rationis, PQ, ad, QΣ, erit vt, LP, ad, Ρβ, quotuplex
ergo eſt, LP, ipſius, Ρβ, radio primi circuli æqualis, totuplex erit,
QP, ipſius, QΣ, eſt autem etiam totuplex, QP, circumferentiæ, C
DFB, ergo, QΣ, erit æqualis circumferentiæ, CDFB, eſt autem, L
P, æqualis ipſi, AB, ergo triangulus, LQΣ, circulo, CDFB, æqua-
lis erit. Dico vlterius trilineum, LQΩ, æquari ſpatio circuli, CD
FB, nempè contento ſub ſpirali, GSIB, & voluta, GB, ſi enim nõ,
erit eo maior, vel minor, ſit primò, maior quantitate ſpatij ſeorſim
expoſiti, 8, diuiſa autem bifariam, QΣ, in, ℟, iungatur, L℟, rur-
ſus bifariam diuidantur, Q℟, RΣ, in punctis, & , Π, & iungantur,
& L, ΠL, & ſic ſemper fiat donec deuentum ſit ad triangulum mi-
norem ſpatio, 8, ſit is triangulus, LΠΣ, per puncta autem, in qui-
bus, LΠ, L℟, L& , ſecant curuam, QΩ, ſcilicet per, O, V, Z, du-
cantur, QP, parallelæ, 7Γ, 69, Φ+, quæ ſi producantur ſecent, β
P, in punctis, 2, 3, 4, quia ergo, Q& , & ℟, ℟Π, ΠΣ, ſunt æqua-
les facilè oſtendemus per Coroll. Prop. 9. huius, etiam, P4, 43, 32,
2β, eſſe æquales, ſimiliter facilè oſtendemus, trapezia, QZ, ZV, V
O, & triangulum, LΟΓ, ſimul collecta æquari triangulo, LΠΣ, . i. eſſe minora ſpatio, 8, habemus ergo ſpatio, LQΩ, circum ſcriptam
figuram ex triangulis, LQ& , L+Z, L9V, LΓΟ, & aliam eidem in-
ſcriptam ex triangulis, LZ+, LV6, LO7, LΩΣ, compoſitam,
quam circumſcripta excedit minori ſpatio, quam ſit, 8, ergo tri-
lineum, LQΩ, excedet inſcriptam multò minori ſpatio, ergo in-
ſcripta erit manior ſpatio, GMSIB, quod eſt abſurdum, nam ſi cen-
tro, A, ſemidiametris æqualibus ipſis, L4, L3, L2, deſcribantur ſe-
ctores, vel ſectorum reſidua. AkIR, XSN, TME, habebimus ſpa-
tio, BISMG, inſcriptam figuram ex ſectoribus, vel ſectorum reſi-
duis iam dictis compoſitam, & aliam circumſcriptam ex ſectori-
bus, vel ſectorum reſidius, BAC, IAR, SAN, MAE, compoſitam,
& quia, ΣQ, ad, QP, eſt vt, βΡ, ad, PL, & , PQ. ad, Q& , eſt vt, L
P, ad, P4, ex æquali, ΣQ, ad, Q& , erit vt, βΡ, ad, P4, ideſt vt GB,
ad, BK, ideſt vt circumferentia, CDFB, ad circumferentiam, CB,
(nam dum punctus, B, deſcribit totam circumferentiam, CDFB,
punctus deſcribens ſpiralem percurrit ipſam, GB, & dum, B, deſcri-
pſit circumferentiam, CB, idem punctus percurrit ipſam, Bκ,) eſt
autem, QΣ, æqualis circumferentiæ, CDFB, ergo, Q& , æqualis
erit circumfer. CB, eſt verò, Q& , ad, ΦΖ, vt, PL, ad, L4, ideſt vt,
BA, ad, Ak, ideſt vt circumferentia, CB, ad circumferentiam, IK,

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