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# Title: [Problema Austriacum Plus Ultra Quadratura Circuli] Problema Avstriacvm Plvs Vltra Qvadratvra Circvli (shelfmark Leibn. Marg. 230, Stück 1)
# Folio 3, page id 1815201908:0060
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oM.
Re
A
OLLHLIIE.
R OPOSITIO IV.
Ecta SIE AB: in C ee D. nedia e extrema ratione propoctionali, de
Dcompoltca ex acesd exemarum; de media; Verbigracia C B, bifa-
OICO L L GuaataC tectangulo D E. A, Xquale exlurgere.
Demonstratio.
DRoduCaTFC A. IM F, VE C A, AF aquales snEinter SC: guOMIAI CAL BE B A 20
AC: ita BD ad DC; eric componendo, B A cum AC. hoc eA BF. ad Ti R.
id est AC, ve BC ad DC. Sed corius BF, dimidia est A F, ipsus quoque BC, di-
midia eAt FC, per constructionem.
Igitur AP, ad AC, CAVE F C ad CD. e conuerrendore AF. ad C P ita est,
EC ad DE. ErBO AF. CE. DF. sune in continuara racione. Vnde _ quadracO, 16. s
CF. rectangulum AP D. aequalc. ODuod Fuic demonstrandum.
P R OP OSIT1O 
DOancureres h ContinuaratIONO F D, F. C, FA. X LB, Hac~qu2-
II LC, _  C A. aqualis AF. dico A B. in C. d D. diuisatm este
media _ extremararione proporrionali.
Demonstratio.
riam seceturin F.
ad AC. Guate d bC da DC. Ita BF. ad AC. hoc ett ad AF. ex diuidendo
BD ad DC. ve B A ad AF. hoc est, ad AC. Ouod tuic demonstrandum:
eain etratiOCON
Lcinuata, Erium FD. L
F C. P A. erit quoque di
nidendo, DP ad Ve. e LC. de CA. lecomone eL al. MA
Corollari.
role Consequum,  ICclatSatam BDC. SCCLAHSHNO AVL. CHC
aequale, allete BA C. rectanoulo aequale exittere, Ii AD cectangiilun.
Primi demonstracio inde parec: quoniam BC. in P. Sectâ est in aequalia, et hioi
aequalia in D. rectangulum lub inx qualibus legmentis cocius BD C. ina Cu 2 qu2- 2 S. Sccun-
deaeo quod sic ab incermedia DF. aequale est ci; quod a dimidia CR. describieur
quadrato. Rurlus cum P. F secta Ac in D. recunque erit rectatiguluim AR D: b ae qua- b 3. Secun-
Lerectansulo ADB, _ quadraco D R. sed iam octensum est, AF D rectangulum 
euadraco CFi aequale, Igitur BDC rectanigulum, una cum DF quadeaco, aequa:
se est AD F rectangulo; vna cum D F quadraro. Sublaro icaque communi D F, qua-
ataro, reliduum erit BD C rectangulum, rectangulO AD F. X qualc:
que addicmrpars C. A, erie rectangulum B R C. cvria cum quadrato C F, Xqualcc 6. Sccun-
quadFaEO A P; sed CuM SCCCFUE A F, in D. VecUngUC, eriE quadeatUm AF, quale ai.
duobus d rectangulis AF D; 8 F: AD. Toicur rectangulum B A C, vna cum quadra- d +. Sccun-
t0 C F, 2qualc est duobus rectangusis AF D; F AD. Sed HOTUNI AIECTUM A F: D, d
octentum eA quadrato CF. essec xquale, his Ieaque derractis, residuutn mancbiee +. Huiu
B A C rectangutum, rectansulo P A D aequalc:
AIECEUM quIOGNE HE MANIFCIEUM CFIE. Suonta BOinF; secta est bitariam; ipsi.