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# Title: [Problema Austriacum Plus Ultra Quadratura Circuli] Problema Avstriacvm Plvs Vltra Qvadratvra Circvli (shelfmark Leibn. Marg. 230, Stück 1)
# Folio 313, page id 1815201908:0370
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H
DRC
E L L IPSIS
PROPOSITIO CXXXIL
Tliplim ABC contingar in B linea BD Condeniens cum axe maio-
sre C A, inD. ex B aucem concactu, normalis ad conringencem po-
LaLUE BF, Occurrens axI INE.
Dico FB lincam breuillimam esse illarum quxe ex L. puncto ad peri-
heriam ellipleos duci poltune.
Demonstratio.
Enero L inceruallo F B circulus describa-
CuF F B C OCCurrenS axi in F _ C. Cen-
erum ellipseos sic II. Ouoniam D B ellipsm
concingens cu axemaiori convenic in D, dt an-
sulus D B F rectus est, BP linca non cran(IE
per II cenerum ellipseos: E enim P. Cencrum
est, recta F B, normalicer ad contingencem po-
lica axis erie coniugacus axi AC, (cum aequi-
distances omnes contingenci DB abitariam ec
ad rectos dividerer.) adeoque DB aequidistarer
Axi AC: non igieur P cencrum cs ellip(cos,
neC P B diameter: quia vero DB cum aXC
connenic ad partes A, F B lineab minor est sc.
midiamerro, sibi parallela: adcoque de minor
est lemiaxe HC. e mulco minorrect a FC, qua-
re circulusradio FB descripeus, occurric axi in
C incra ellipum, punctum ioitur , Cupra C eAt.
Rurlum cum ETCid es R FI, maior licosten-
sa quam P, B id est P. C, ablaco communi F. FI,
MANCE A E maior quam FTC, POSICA AUTCM FF
aequali D HI, recta F L XquIaTUE FIC, isitur AF quoque maior est FI: ablato ex-
8O Cx F E _ AF. communi IF, manec I A maior quam IF: vnde e L punctum
cadic inera elliplim intra A. vIceriuS Bonatur Per F, contingens FR, cui aequidistet
PONM, CTILISITUE DE MB quadracum ad quadratum BK (IC CO MO rectansu- C35. terrus.
lum ad quadratum F K. sed re M B quadracum ad quadracum BK d SIC PNINT re
ctangulum 2d rectangulum a R B; igicur ve O MO rectangulum ad quadeacum
FK lic PMN rectanoulum est ad rectangulum a RB: _x permutando, inverten-
dove PK quadratum ad rectangulum a KB, EC OMO, rectangulum est ad re-
99. hacius
ctangulum PMN: est autem F K quadracum maius rectangulO C aKB, igitur _ C6. sccundi
MO rectangulum maius est rectangulo PMN: icerum OMO rectangulum
vira cum quadraco ZO aequalc est quadraEO ZMI, _ PMN rectangulum una cum
quadraco Z. N, eidem quadraro ZM aequale cst; aequale isicur est rectangulum
DMO arnicum quadraco Z O, rectangulo PMN, una cum quadrato Z. PI; a
quibus si maequalia auferancur rectangula OMO, PMN, inaequalia FCMAMEHE
quadrata ZO, ZN: ∞x quia OMO rectangulum manis est rectangulo PMH.
quadracum ZO minus CsF quAdFATO Z N: X ZO minus quadraco P Z; puncta igi-
cuE O _ O inera ellipsim sime: similicer ostendeneur puncta Z F, et quaeuis alia
perimerri circuli FB C eseinera elliplins circulusigieur FB C eorus intra ellipum
codic: unde cumrecte omnes ex Fi contro circuli ad elliplis peripheriam ducte prius
circulo occurrane quam ellipli: adcoque semidiameeris eiuldem maiores linc: igi-
TUF EB, quae in B puncto communi cllipsi _e circulo cerminatur omnium illarum
brenictima cA quae ex F. puncto ad elliplis peripheriam duci poslune. Ouod erat
demonstrandum.
2 13. huins
10. huinr