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# Title: [Problema Austriacum Plus Ultra Quadratura Circuli] Problema Avstriacvm Plvs Vltra Qvadratvra Circvli (shelfmark Leibn. Marg. 230, Stück 1)
# Folio 213, page id 1815201908:0270
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to
CTRCVErS
P R O POSITIO LXLAYI.
Ccurrar icerum circulus ABC, cuins diametet AC, ClMTO CTTCuM
AF. C, ductaque ex A recta A B F, ponacurnormalis BO Lad AC.
Dico lincas AC, AF, e compolicam ex AC ee BF, in continua cf-
cadalogia.
Demonstratio.
Vadracum AF, xe quacur quadraris AB, BF, et
A BF a cectangulo bis sumpco: cst aucem qua-
dracum B F aquale B C b quadraEO, eC AB F rC-
etangulum bis lumpeum aequale AB C rectangulo
(hoc CAC AC. BD) bis sumpco, hoc est rectangulo
ACBF Cemel lumpeo; igicur quadratum AF, X-
quale est quadeacis A B, B C, hoc est quadraco AC.
e rectagulo AC, BF, lemel sumpeo. Sed AC, qua-
dracorna cum rectangulo A C B L, aeguatur rectan-
Bulum Cuper AC _ compolica Cx AC, BF; 1gi-
cur quadratum AF, aequale est rectansulo (uper
A C de compolica ex ACBF. Vnde linca AC,
A E _ COmpOLICA CX A C B F in contina luncana.
logra. Ouod Fuic demonstrandum.
PROPOSIFIO LXZXVIL
Xera aream circuli A B C, sumpeo puncto E, demMacut ad alaHe-
Lerum AC normalis F. D, Ponaturque quaenis alia F. IK.
Dico (i D punctum communis lincarum AC, D F: incerlectio cadac
intracirculum, quod E. D quadratum superet rectangulum IF. K re-
ctangulo ADC: sivero D, extra cadat, dico quod E. D quadratum detr-
ciae a rectangulo lF. K, rectansulo ADC.
Demonstratio.
DRoducta D E, circuli perimetro occurrat in B: qua-
dracum DE aquale est quadraris DC, GF vna
cum DCF rectangulo bis sumpro id cst reccangulo
BO E Cemel sumpeo. sed BCF. rectangulum vna cum
quadrato CF. aquatur rectangulo BF C, igieur
quadratum D F. aequale cA quadraro DC, id cA rc.
etanguloC D Arna cu reccangulo GF B id CAIF R.
Ouod HD exera circuli arcam occurrar diamerro
AC prOdUCEX, AUICAEUF CK E FCCTA F A, OCCUETeNS
perimerro circuli in F, iungancurgue CF: cruncica
que similia criangula ADF, _ C AF, QDA, AF.
lacera proporcionalia laceribus AF, AD: vnde _ re-
crangula C AD, F AF, lunc incer le xequalia; quia
VerO QUadraTUM A P aquacur recranoulis A R F, F A F, cidemgue AF gnadrato
xquantur quadraca A D, D Esquadracum aurem AD, aquale cAt reccangulis ADC,
DAC, cruncroctangula AP, F, E A F, aequalia recrangulis AD C. D A C vna cum
quadraco DF, octensum aurem est reciangulum D AC, recrangulo LAF. aquale,
quare relidua quoque inter le xqualia sune, id cA CD A reccansulum auctum qua
draco DF, aequale reccangulo YP. A, id eA LV. K. Guod erac demonstrandum.
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