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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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SP. H A R OFDPS
S
quibus inscribaneur parabole ADC, D FF.
Dicolphaeroides AB C ad sphaerolees DF F habere rationem com-
policam ex parabola ABC ad D E L patabolam, e ex rarione alcicudi-
nis B e ad F. HL alticudinem.
CX2
X
TINE ABC, DEF sectiones Factae per ares sphaeroidcon ABC, DP P.
Demonstratio.
aSc enim ellipcicum planum A BC du-
ctum in sc nd DP F plavum ellipcieum
in se ductum recorpushabens balim ABC
Parabolam de alcicudinem B C, ad Tolidum
Tuper BaE DF F parabolica e alcicudine
BFI hac enim illis aequalia esse Facile cor.
ligetur cx 204. libri de parabola, e
45. libei de ductibus. Ced Ecue se habet
AB C, ellipricum in sc ductum ad D F. F
ellipricum ductum in (C, ita quoque AB C
ductum in orbem ad DE. F, ellipeicum du.
ctum in orbem, cum quadraca IR ad L NI
eandem racionem concineane quam circuli
radiis illis delcripei. Igicur eriam sphaeroi-
des ABC adDF F, cam haberracionem
quam parabola AB C ducta in alcicudinem
BC, ad parabolam DF F ductam in alci-
cudinem F FT. Ouod tuie demonstrandum.
Verum quoniam argumencandi racio sub
finem thruius demonstracionis vlurpaca, non
adco Forrasle plana omnibus ridebitur, ve
vlceriorem probarionem non deliderene,
adiungo lemma vniversale huiusmodi.
Vaca linc quaenis plana ABC, DL. F. Di.
co lolida quae hune ex planis illis in (c du-
etis candem incer serationem habere, quam
Colida que proecCaneur ex ijidem planis ductis in orbem.
Ducantur in veroque plano parallelae intinicae, IK, I. M. _e concipianeur supeE
guadracis TK, IM consticui paralle lepipeda infinira in alcicudinibus KK, MM. Ma-
nifectum esthaec parallelepipeda inleripra esse Colidis, qua hunc ex ductibus plano-
rum ABC, DF. F, in se. incelligantur deinde radiis IK, de L MI deseribi circuliin-
tnici, ac super ijs crigi exlindri infinici in alcicudinibus ij (dem KK. MI M. PatechosCy.
lindrosinseripeos esse lolidis, quae hunc explanis ABC, D I. Ei duccis in orbem: lam
vero quoniam parallelepipeda de exlindri habene communes alcicudines; crunc in-
cersevebales, scilicee ve quadrara IK ad circulos radiorum IK. led quadraca sinou-
la IK sunc Cingulis circulis radiorum IK proporcionalia. ergo etiam parallelepi.
peda lingula Cunc Engulis exlindris proporcionalia. Fr∞o 12. 5. vernum parallelepi.
Dedum IK ad primum exlindrum PK, ica omnia parallelepipeda ad omnes cylin-
deos. pari argumento vernum parallelepipedum L M ad rnum cxlindrum LM,
ita ommia parallelepipeda LM 2d omnes cxlindeos L. M. sed vernum parallelepi-
Dedum TK, ad unum cxlindrum TK., ica rnum parallelepipedum P. MC ad ruum
cxlindrum I.M, cum horum quaruor Colidorum rario eadem sic qua basium, quas
Pacec esseproportionales. Ergo omnia parallelepipeda TK, lunc 2d omnes cylin-
dros FK, rcomnis parallelepipeda L M ad omnes cylindros L MI, ex permurando.
Xc. Vnde tacile ium probabitur iuxca Formam dilcursus libri de ductu plani in pla-
num,