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# Title: [Problema Austriacum Plus Ultra Quadratura Circuli] Problema Avstriacvm Plvs Vltra Qvadratvra Circvli (shelfmark Leibn. Marg. 230, Stück 1)
# Folio 1215, page id 1815201908:1272
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QKADRAEYRA H ZPERBOLA
Demonstratio
Sensum est CupFA ATaM LECLANGUIa LMEN, OEP, RM H quam rectanouta a 12q. Hi
YRS, X R, CR L Candem concinuarerationem quare cum extrema rectaM- in-
suta RM FI, OR I hiric aGlalia, erio racio rectanguli LM N 2d V R S, roctangu-
lum duplicaca bracionis, quam habes rectansulu, OM P 2d XR F sect ansulum. b g. tinim.
Duoderac demonstrandum.
lRC1031IIOCALAVI
IIdem maHCHCDUS:
Dicorectansuum OA e ad XR. P, duplicatam racionem habere
eius quam habet tectangulum BM O ad YRF.
Demonstratio.
Scensum estc Tam reCtaL BCIA OMLEs BMEO RMEI, quam rectanouta Z R Y curs. huius
RF e CRL in eadem ese proporrionc. quare cum rectangula postrema
XMHT: CR L incerle sine xqualia, eric drario rectanguli OMP ad X R T, rectan. 83. buiuc.
pulum dupie acaracionis rectanguli BN O ad YR E rectangulum.
PROPOSITIO CXXXVIII
encibus denuo ijidem:
Dicorctangulum L MI N ad rectangulum V R S quadruplica-
tam eius continere rarionem, quam haber rectangulum BMI O ad Z R F.
Demonstratio.
Eatangulum LMIN ad VH rectangulum, duplicacam chabee ejusrarionem, c 2S. hN-
quam haber rectangulum M P ad. Z R F rectangulum. Sed OMP rectangu- ins.
dum de rectangulum Z R I. Haber Fduplicatam eius quam rectangulum BM O ad E1y7. Ln
YRF rectangulum. ergo L.M N rectangulum ad VRS quadruplicacam habee ins.
cins rationem quam rectangulum BM O ad Y R P.. Quod Fuic demonstrandum.
rectangulum.
P Oe C
5
T1O CXXXIX.
Dolcisiildem
Dico rationem rectanguli L. M L A SCCLAHSMIAM V RS, tocIeS
concinere permulciplicacionem lallolle ectanguli OMPaa XK l re-
Clatgulum, quories tiaec caderatio OM P ad Z R T continer per mul-
iplcattoeao eclaHB M BM O ad YR L.
Demonuram.
iX praecedentibus est manifesta: nam oHtCM_I CM SLALIOHetm FecLanguli L. MI N ad 0
b.l
TRS duplicaram esse eius, quam habet rectangulum OM P ad Z R T, 2C PrO. Ius.
 prior rario lecudam bis concinceper mulciplicatione. Similicer eA demonstrac
teclangulum OmP ad LRI. Mlpdcacam habereracionem eius quam rectan. s.
gulum BM O ad YR F. rectangulum e consequencer bis continee per mulciplic, tus
Cactol caliolecladal. lale Datet Tactoctu ccclaii e2. ad 13.
cocles concinere per mulpaconactonenea nl al 221C23940.
cieshac concinerrationem rectansuli B AXO nd L K L. rectangulum.
   iuui  e p pe tu t e  es se et e tie e    o  t   e  u ie  t  e  1 1 es e l 1le