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# Title: [Problema Austriacum Plus Ultra Quadratura Circuli] Problema Avstriacvm Plvs Vltra Qvadratvra Circvli (shelfmark Leibn. Marg. 230, Stück 1)
# Folio 66, page id 1815201908:0123
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( u  prm pu . r vur m u 1
PR OGR E SSTON F S
Pappo lib. cerrio propolicione decimaleptima in cribus continue proporrionalibus
dum ad ampliorem rlum.
propolica fuic: guxe cum vnivertalis sc, censuimus eriam hoc iplum verbo indican.
L
7
IE A aequalis B, e C xqualis D; omnibusa A, BBC, D, P
cur aequalis F; duabus vero B, D, ponatur equalis t: limilicet clabus
CD, aequalis C; _ rectae D, xqualis FI.
Dicorationem L ad F, _ C, ad II esse duplam. Duod ( rectisF, F, C, Ll,
aequalis ponaturl, ee duabus Fd HI, a qualts Hacuacu B, ex lincis 6, 11
aequalis ponaturL, denique ea L, at M aGualis;
Dicorationem lad R, e L ad M eNecriplam.
Demonsblalio.
cum B, dupla est ipsusB; ex C cu D, iphius D dupla est: eroo criam A, B, C, D si-
Cmul sumpex, duarum B, D, Cmul sumprarum duplxe sunc. Sed lineis A, B, C, D
aequalis est F, _ lineis B, D, aequalis F, ergo F dupla est F: Similicer C, D, limul
sumpexe, hoc est G, ipsus D, id est FI, sinc dupla. Ouoderac primum. Secundam
Parcem ita expedimus. E continet rectam A bis, _ C bis; ipIa verO F ContineE A Sc-
inel, _ C lemel: ipla candem C continer C bis; _x TI xequalis est C; igITUE OmneS F, F.
Cn FLhoc es recta I, continer A corrio, e rectam C (exies: recta verd R, COnTIHCE CX
Bypothesi F, _x FI; (ed F continee B e D, hoc est ipsam A Semel, et lemel rectam C,
igitur si addacur II, hoc est C, recta K eric lemel A, ex C bis sumpram. Sed A sum-
Pta Cemel, cum C bis, eAt certia pars A linex rer Cumpexe, e linee C sexies (IMPEX:
Facio ergo I ad K cripla cst. Fodem modo ostendemus quod L, Sic cripla linceM,
nam I aequalis cA C, II; ipla vero C concineebis C lineam, cui si adiunoaturHI,
ips C aequalis, cric C cum FI, hoc est L, cripla recta C. hoc cstFI, hoc cAt M:
Pacec ergo veriras propolrione expolta.
Scholion.
appus lib. 3. proposicione 18. tribus continuis quantitatibus applicuit hanc materiam, quae scili.
cetcandem continuant rationem. quia veroddueriinon solum continui rationidus conne-
nire hanc proportionalium propriecatem, verum etiam discretissimis, modo rationes simtles ast
sumancur: hinc opera presiumduxi, hoc insinuare. Nocaru aurem dienum existimo, se quis v1.
terius
M  ee e  1 .    1e 1.        a  e .  pet su  u  e   e   u     n    a u u  upu