461441LIBER VI.
eumſcriptæ ſuper inſcriptam ſunttrapezia, HL, LK, KI, cum tri-
angulo, IFQ, quę, quia ęquantur trapezijs, ΓV, VN, NQ, &
triangulo, IFQ, (nam dicta trapezia ſunt reſidua triangulorum in
ęqualibus baſibus, & altitudinibus conſtitutorum) ideſt triangulo,
FΓG, ſubinde ſunt minora ſpatio, Ω, & ideo circumſcriptã ſuperat
inſcriptaminori ſpatio, quam ſit Ω, ergo trilineum, FLHG, excedit
inſcriptã multò minori ſpatio, excedit autem ſpatium reſiduum cir-
culi, ECDB, iam dictum ſpatio Ω, ergo ſigura inſcriptaerit maior
dicto ſpatio reſiduo; quod ſerua.
angulo, IFQ, quę, quia ęquantur trapezijs, ΓV, VN, NQ, &
triangulo, IFQ, (nam dicta trapezia ſunt reſidua triangulorum in
ęqualibus baſibus, & altitudinibus conſtitutorum) ideſt triangulo,
FΓG, ſubinde ſunt minora ſpatio, Ω, & ideo circumſcriptã ſuperat
inſcriptaminori ſpatio, quam ſit Ω, ergo trilineum, FLHG, excedit
inſcriptã multò minori ſpatio, excedit autem ſpatium reſiduum cir-
culi, ECDB, iam dictum ſpatio Ω, ergo ſigura inſcriptaerit maior
dicto ſpatio reſiduo; quod ſerua.
Diuidatut nunc, AB, ſim liter, ac diuiditut, FG, in punctis, 3,
4, 7, centro autem communi, A, ad diſtantiam punctorum, 3, 4,
7, deſcribantur circumferentiæ, 35Δ, 4ΣRβ, 789M, ſecantes ſpi-
ralem in punctis, S, R, M, per quæ tranſeant eductæ à centro, A,
productæque vſque ad circumferentiam, ECDB, rectæ, AD, AC,
AE, vt igitur in præhabita demonſtratione oſtendemus circumfer.
53, & rectam, IQ, inter ſe æquales eſſe, & ſimiliter circumferen-
tiam, RΣ4, æquari rectæ, KT, & , M987, ipſi, LV, & quia, 53,
11Corol. 1.
3.huius. circumferentia ad circumfer. Σ4, eſt, vt, 3 A, ad, A4, ideſt vt, QF,
ad, FT, ideſt vt, IQ, ad, NT, eſt autem ęqualis, 53, ipſi, IQ, ergo,
Σ4, erit ęqualis ipſi, NT, & eſt, 34, ęqualis ipſi, QT, ergo faſcia, 53
4Σ, erit ęqualis trapezio, IQTN; eodem modo oſtendemus faſciã,
R9874, æquari trapezio, KV, & faſciam, MECDB7, ęquari tra-
pexio, PLVG, & ideo figura compoſita ex dictis faſcijs ęqualis
erit figurę compoſitę ex his trapezijs inſcriptę trilineo, FLHG, eſt
autem hæc figura inſcripta maior ſpatio reſiduo circuli, ECDB, ab
eo dempto ſpatio ſub ſpirali, & voluta, AB, ergo figura compo-
ſita ex dictis ſpatijs erit maior ſpatio dicto reſiduo, cui tamen eſt
inſcripta, quod eſt abſurdum, non ergo trilineum, FLHG, maius
eſt dicto reſiduo.
4, 7, centro autem communi, A, ad diſtantiam punctorum, 3, 4,
7, deſcribantur circumferentiæ, 35Δ, 4ΣRβ, 789M, ſecantes ſpi-
ralem in punctis, S, R, M, per quæ tranſeant eductæ à centro, A,
productæque vſque ad circumferentiam, ECDB, rectæ, AD, AC,
AE, vt igitur in præhabita demonſtratione oſtendemus circumfer.
53, & rectam, IQ, inter ſe æquales eſſe, & ſimiliter circumferen-
tiam, RΣ4, æquari rectæ, KT, & , M987, ipſi, LV, & quia, 53,
11Corol. 1.
3.huius. circumferentia ad circumfer. Σ4, eſt, vt, 3 A, ad, A4, ideſt vt, QF,
ad, FT, ideſt vt, IQ, ad, NT, eſt autem ęqualis, 53, ipſi, IQ, ergo,
Σ4, erit ęqualis ipſi, NT, & eſt, 34, ęqualis ipſi, QT, ergo faſcia, 53
4Σ, erit ęqualis trapezio, IQTN; eodem modo oſtendemus faſciã,
R9874, æquari trapezio, KV, & faſciam, MECDB7, ęquari tra-
pexio, PLVG, & ideo figura compoſita ex dictis faſcijs ęqualis
erit figurę compoſitę ex his trapezijs inſcriptę trilineo, FLHG, eſt
autem hæc figura inſcripta maior ſpatio reſiduo circuli, ECDB, ab
eo dempto ſpatio ſub ſpirali, & voluta, AB, ergo figura compo-
ſita ex dictis ſpatijs erit maior ſpatio dicto reſiduo, cui tamen eſt
inſcripta, quod eſt abſurdum, non ergo trilineum, FLHG, maius
eſt dicto reſiduo.
Dico neq;
eſſe minus.
Sit, ſi fieri poteſt, minus ſpatio eodem,
Ω, ſit autem vt ſupra trilineo, FLHG, circumſcripta figura, ex
trapezijs, KQ, LT, HV, & triangulo, IFk, compoſita, & alia ei-
dem in cripta ex trapezijs, PO, OT, NQ, ita vt earum differentia
ſit minor ſpatio, Ω, igitur circumſcripta excedet trilineum, FLH
G, multò minori ſpatio, ergo circumſcripta figura minor erit ſpa-
tio reſiduo jam dicto circuli, ECDB, quod excedit trilineum, FLH
G, ſpatio, Ω, quod tamem eſt abſurdum, nam ſectorem, AS3, pa-
ret ęqualem eſſe triangulo, FIQ, faſciamque, ΑRΣ43, æquari oſtẽ
dem us trapezio, kQ, modo ſupra adhibito, & faſciam, βΜ874, ip-
ſi trapezio, LT, & totam faſciam, 679C, trapezio, HV, vnde fi-
gura compoſita ex dictis faſcijs, & ſectore, A53, erit ęqualis
Ω, ſit autem vt ſupra trilineo, FLHG, circumſcripta figura, ex
trapezijs, KQ, LT, HV, & triangulo, IFk, compoſita, & alia ei-
dem in cripta ex trapezijs, PO, OT, NQ, ita vt earum differentia
ſit minor ſpatio, Ω, igitur circumſcripta excedet trilineum, FLH
G, multò minori ſpatio, ergo circumſcripta figura minor erit ſpa-
tio reſiduo jam dicto circuli, ECDB, quod excedit trilineum, FLH
G, ſpatio, Ω, quod tamem eſt abſurdum, nam ſectorem, AS3, pa-
ret ęqualem eſſe triangulo, FIQ, faſciamque, ΑRΣ43, æquari oſtẽ
dem us trapezio, kQ, modo ſupra adhibito, & faſciam, βΜ874, ip-
ſi trapezio, LT, & totam faſciam, 679C, trapezio, HV, vnde fi-
gura compoſita ex dictis faſcijs, & ſectore, A53, erit ęqualis