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# Title: [Problema Austriacum Plus Ultra Quadratura Circuli] Problema Avstriacvm Plvs Vltra Qvadratvra Circvli (shelfmark Leibn. Marg. 230, Stück 1)
# Folio 486, page id 1815201908:0543
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41.
b 262. lu.
d262. h-
P A R ABOL A.
Demonstratio.
DOnancur ordinatim C A, FD: VEBFF ad BC, AICF K
est ad F. F, isicur re quadracum OTIad quadrarum AC.
sic IK. quadratum ad quadratum DF, X invertendo
permucando ve AC quadracum ad quadracum DP, Sc
CHI quadrarum ad quadrarum PK, _ VE AC adD F, SC
GFT ad IK. scd AB C parabola est ad parabolam CBH
a in eriplicaca racione AC ad CHL; e DFF parabola
ad parabolam PF. K, in eriplicaca racione D F ad FR, crie
isicur reparabola AB C. ad parabolam CB FP, sICD R Li
parabola ad parabolam IF. K: X permucando vE ABC
Parabola ad parabolam DF F, Sic C BFI Parabola ad Pd.
rabolam PF R; quod erac demonstrandum.
Si vero BC, F F laceribus rectis XgUeNEuE, Crit CBHI
Parabola ad parabolam PP. R in duplicata rarione CHI
ad TK: quia ABC parabolaad parabolam D P. Fiin du-
plicaca estratione AC ad DF, cum AC, C B linex, icem
DF, FF ex quibus brationem habene compolicams
aequales pOnanEUE.
113C
DR BEOSTTIO CCLXXXI.
proporrionales sine, poltisque Ordinarim E G, FB, D C ex C _ B,
diamceri demirrantur C HI, BL Ocurenres F B, D C lineis in HI X I: e
pet F, II X F, L parabola descrrasseur habences apices in F X F.
Dico FF LI parabolam elle ad patabolam D FI, in criplicara racione
Demonstratio.
Voniam A E AF, AD concinue proporcionales
VE AF adC AF, AC F, F est ad FD: sed ve
AF ad AF, SC LC quadratum eA ad quadracum
FB, id est quadracum F FT ad quadcacum D I; ioiene
De P F ad F D, SC FIT quadracum est ad quadracum
DI: quia vero ratio parabola FP. FI ad parabolam
D FTcompolica cltCx drarione B F ad FiD, id est
ex duplicata racione LiI ad DP, d icerum exra
tione F TTad D I, Parabola F F. TT cs ad parabolam
D F I, in criplicala racione FFI ad D I. Quod crat
Seo A B C parabolae aris A D divisus in F _ F, vE AF, A F, AD
tHIad DI.
ossionibi.
CI. DEpro
demonctrandum.
Corollarium.
Tinc sequicur, sipuram mixcilineam FTCFB, CHC aet HIV MLAIH ABC III CIPIICala
racione IIB ad IC. cum enim F CF, FB, DC proportionales sinc, parabolae
241: D A G, F AB, D A C in cconcinua quoque sunc analogia: quarC _E F E C B mixti-
El. Depro-
lincum ad mixcilineum DFBC cIL AC PalaVA au ParabOlam F AB, id elt
pressionibus. in criplicaca ratione P. C ad F B, I CH F HL dd D L. Ted _ LCCaHGutuM FC, ad re.
Bnius.
etangulum DB est in criplicata rarione linex F HI ad lineam DF, quia rationem
adresiduum LB C. in criplicata eA rarione LEl ad DL. quia vero ellC, L. ad LB. 1d ell
FH ad
habent compolicam ex ratione F F ad FD, _x F FT ad D I. igitur ex reliduum HCB