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# Title: [Problema Austriacum Plus Ultra Quadratura Circuli] Problema Avstriacvm Plvs Vltra Qvadratvra Circvli (shelfmark Leibn. Marg. 230, Stück 1)
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ELL IPSTS
PROPOSITIO LXXIIF
sencibus aequale cM parallelogramito concenco lineis exrrema qua-
rumuis Matecto Comuhalau Comungeneibus.
Demonstratio.
InE ABC ellipseos AXeS ABCD, ex alia quauls
diametrorum comiusacio P F G FI Tungancurquetam
2xium, quam diamecrorum exerema: dico C AD B Pa-
rallelogrammum aequari parallelogrammo P, IT F C. rC-
ctangulum ex A B _ D I duplum est crianguli AD B:
de rectangulum ck AB _ CI, duplum est crianguli
ACB. ergorectangulum ex AB, CD, duplum est p2-
rallelogrammi ACBD, Sinc A C B D parallelograM.
mum, dimidium est rectanguli super AB, CD. limili-
cer octendam patallelosrammum F. C F. II dimidiun
esse parallelogrammi (uper F F, CFI in angulo FIFI:
sed parallelogrannum super RL, CFT. xquale cst pa-
rallelorrammo supet A B CD; igitur _E AC B D paral-
leloprammum aequale est parallelosrammo P. C FH.
Demonstratio.
Vm exhyporhek DE sic ad E. B vE A F ad F C, exie
permurando DH ad AF, VE BF 2d FC. quare
SEFOTA DB, ad EOFAM AC, VE DF ad AF, _E F B, 2d
FC. ioicurrationes BC, ad F C, _E DF ad AF, simul
sumpre duplicaex sure racionis D B ad A C. AEqui ra-
tio rectanguli BF D del rectangulum AFC, compo-
nitur exracionibus BL 2d F C, _ DF ad AF. CrgO
rarto rectanguli BF. D dd rectangulum A F C duplica
ea estracionis BD, ad A C, hoc est rationis BI ad AL
ergo lectanguluI DL. D est ad rectangulum A F C, ve
quadracum BI ad quadracum AI. Ccd idem quoquere-
atangulum BF. D est ad quadracum CE, ve rectangulu
N ellipsi paralleloerammum quod hc à lineis excrema axium coniun-
Dubature demonstrandum.
ROPOSITTO LZZIV
VInE ABC ellipleos axes vel dianerri coniugatae, AC, BD. diuise
que BD vccunque in L, diuidatur AC in F proporrionalicer, de per
F. _ F ordinarim ducantur linex L. (1, F II:
quadrato HIF, e liguaatacum CF. lic aequale rectansulo AF C. dico
Dicorectangulum A F C aequari quadrato C F, et B F. D rectangulum
BD, AC proporcionalicer esse dinilas in F. _ F.
EIL xqualia csc.
Sineiam aequalia quadrarum C+F. _ rCclaHOIum AF C: dico BD, AC pFOPOr.
cionalirer esse sectaS: Nam si non eA ve B F 2d E D, sIC AF ad F C, SICVE BK, 2d
BFD, hoc est, quadratum B I, ad quadratum A I; Xquancur igitur quadracum C L
de lectangulum H C.  nmus rectangulum BF. D, _ quadrarum
172. huiuso
Iam
ED, GC AR dl LC. L C a L de rectansulo AK C per pri