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# Title: [Problema Austriacum Plus Ultra Quadratura Circuli] Problema Avstriacvm Plvs Vltra Qvadratvra Circvli (shelfmark Leibn. Marg. 230, Stück 1)
# Folio 117, page id 1815201908:0174
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tet
eMt
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ON
Len
SH eriE D A ad F A, VEF A ad G A; ac proinde D F eric a ad GF, vE D A 2d a 1bid.
L. Duare cum D A ipliuS F. A dupla SIE, Ctraim D F duxla eric P. C. vnde CL
Glladcupla est ipsius P. C. Similicer ostendemus F. C duplam eAe PHI, iplam aurem
E duplam eAe GL, proindeque F C ipsius CL quadruplam ese: aIguC ita CON-
cinuando line caru, per alcernam illam bilectionem consticui proorectionem maoni-
cudinum CF, F, C, C+ L, ec. proporcionis quadrupla. eodem aurem discurlu quo
priusrltuimus, demonstr abiinus DF esse quadruplam PHI, d FHI quadruplam
sequencis cermini, ac proin de etiam hic progressione rationis quadeupla Cacui. VIce-
rius quoniam cam ratio CB, ad D B, quam IB ad K B, dupla est, erir C B 2d DB, vC
IB ad KB, _b CI 2d DR, vE IB ad KB. Teaque cum IB dupla sIC R B, ctiam brbid.
C. I iphus DR dupla eric; similicer DK iptius F.I duplam esse demonstrabimus.
Psitur CI quadeupla est R L: ideoque Cl est ad P L, ve C F ad P G. vnde pro-
srectionis CE C P. C exC. terminus est I; eadem mechodo discurrendi, ostendecur c73. huiusi
DK CAOZd FK, vE DF CA ad F FI. OuarO _ progrelsionis DF, FFF, _C. cermi.
nuSeriE R. dum igicur reraque progressio C F, F. G, _C. DP, F FT, _cc. constiCAEuE
ab alcerna illa bilectione, in propoticione propolica, ipsius quoque termini crunc in Y
e Kirbicritariam diuiditur magniendo AB. Ouoderac demonstrandum.
Sssumpsimus A C minorem aut maiorem terria parre magnicudinis datae AB; quia si
aequalis uni tercia foret, bisectiones alcerna in eadem semper puneta L es K inciderent; vii
manifestum est; assercionem proposicionis consideranii.
Lemma
e ARS PR IM A.
Aca sc magnicudo A B secta in P ec K secundum rationem V ad Z: ILL BC A R
AEad RB, ve BI ad I A. divisa SE deinde A B adhuc aliceF intOF A d LM C
Dicosi C B dividacur in racione V ad X, sectionem heri vIera K in V, item (
vidatur in cadom rarione, lectionem conrinsereincer K et D in F; ex si F. A, sectionem
Fore incer Lec P in C. aeque ica in intinicum.
D A dicac  Lactene au X MCOCI CadCTCMETA LIH F: TMIIaI A AB di-
Demonstratio
M
Vm ARSiEAdK B, vEB Fad J A, erie comporendo AB ad KB, ve AB ad LA: ex-
B0 RB, IA, addicoque communs IR etiaI AR, IB aequantur: vnde cum AR
AC adR A, VEV ad Z, veique TB ad R B, in eadem ratione crie: quare C B (ex daris
maiol quam IB) maiorem haber racionem ad K B, quam V ad Z: erga lectio ipsius
CB, in raciene Vad Z, cadic vlera K in D: LimILIECT CUM A R SIC ad K B, id eA I A. liCuE
V cltad X, eric D A ad eandem P A, in minori ratIOnO, quam V ad Z: vnde sectio
ipsius D AMacOnCV ad X, cadec rlera lin F: Ouod aucem sectio ipluS F B cad AC
olera K. Codem edo prius modoostendetur. HcM quod Mctd D. Vetas B lc oltendo:
Facta C. B at DB, in cacione Y act X. eric P. B adt D P in min oricacione quam V ad. H.
ed Ed