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# Title: [Problema Austriacum Plus Ultra Quadratura Circuli] Problema Avstriacvm Plvs Vltra Qvadratvra Circvli (shelfmark Leibn. Marg. 230, Stück 1)
# Folio 1069, page id 1815201908:1126
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EONOTDES PARABIOLTONH
Demonstratio.
COrmetur parallelexipedum ex ductu
LRBHF, in F I. DHI; habebic illud ba-
sIM REHIF, de alcicudinem FI. Cimilicer
si construncur parallelepipedum ex rectan-
suto MBC C in BN A C, habchichocba.
sIm BC CM et alcicudinem GC. Igieur
hae magnicudines composicam habenE ea-
tionem exrarione balis MBCC ad balim
XPHE, hoc est ex racione crianguli AB C
deriangulum D P. HI, ex ex racione alcicu-
dinis CE ad alcirudinem LI. Scd MBC.
ductum in BN AC aquacur Colidis, qua
hunc ex ductu plani MBOC in planum
COBN a simul cum solido quod AC ex
ductu BO CC in planum B R AC: G-
mili pacto magnirudo ex ductu R P II F
in E L D HI, aquacur solidis qua 0.
riuneur ex ductu plani RB O F in pla-
num FO B. L D, _ ex ductu plani FOFHI
in BEDP. Dividancur iam B G, F FI in
parces aequales in punctis X, Z. Ec quia x
quaneur OB, HIF, eciam X B, Z. L. pares c.
FUNE: AUCANEUTQ; MO Z R S, _ TOZPV
Parallela ad C A, P D. quadrarum F D (hoc
est ex hyporhesi quadracUm C A) est ad
quadracum OP, ve HIF linea ad Z F3 0i
hoc est, ve GB ad ZBs hoc est veis
rurtus quadracum C A ad quadracum  R. Xquantur igieur quadrara OP, OR.
ergO reCX OP, OR, adeoque e OZ, O Z aqualeS TUNC. TUNE VerO _C MS.
PV adeoque _c MX , TZ aequales. dempris ergo aequalibus L, X.
religue TO, NIO aquales (IMIE: _ quia cOCX M S, T V, aequales, eriam OS,OV
aquales erunc. rectansula igieur M OS aequancur rectangulis TOV, RquanEUF
aurem eC quadrAEA O P, O R. ergo omnia rectangula M O S sunc ad omnia quadra-
ea OZ, vcomniarectangula YO V ad omnia quadrata OZ. ErooMCOB du-
ctum in BOCAN, es ad BO CG In BR AG, VEREOR in EOFDL, ad
E OF EI in F P D FI. ErOOMCOBin BOC AN, vna cum BOCG, in BR AC
(hioc veia ostendi, para llelepipedu Cx M C in CN) est adM CO B in BO C AB, vE
RFOE iN FOFD L, Vna CuM E OFTI, in F P D FT (hoc est ve parallelepipedum
ex KII, in FTL) cA Ad RF O B in B OFD L. igicur permucando vecorum paralle
pipedum exM C:I CNV, cA ad torum parallelePIBCdUM CxK FT in FTL, ica ablacum
MCaB in BOC AI, ad ablacum KEOF, in FOFDI. ergod reliquum
BOCO in BR AC, est ad reliquum F OF FI in F PBFT, reparallelexipedum
ad paralelepipedu, hoc cs (ve ostedi supta) in racione composica ex racione crianguli
ABC ad criangulu D P. LI, _ exracione ipsius A C, ad F. I. Duod erar demonstrandu.
Scholion.
s. C
emor jum Hord ac parabcta demonsratum esse, nullas darr parabolas nclinatas, quod
sensui huius proposirionis videtur repurna:e: nam si mulriplicario parabolae BC C in
BCA, per ductum facta qui quadraca producit fiat circulariter, e comparetur productum
cum marnitudine FHF, quae etiam muliiplicatione circulari formabis aliquod solidum, hune
vera fiet comparario conoideos cum conoide, nam B  C C ductum in orbem producit verum
conoides; undeeriam P. P. II ductum obliqua inclinatione in orbem exhibebit inclinatum co-
noides, nibilominus verum que subsistis, recte scilicet affirmatur non dariparabolas inclinatas,
o te  per se tet  tt at stu et u t at tat  tat totu et tete   ett   t e  t t  t t te pett at  tete t tet s t     pti et te  pi t  t    u eate  s itisi