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# Title: [Problema Austriacum Plus Ultra Quadratura Circuli] Problema Avstriacvm Plvs Vltra Qvadratvra Circvli (shelfmark Leibn. Marg. 230, Stück 1)
# Folio 86, page id 1815201908:0143
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  t uu pu u    HHI
Ar Sexti.
bIbid.
P R O GR F S S FO N F 5
Demonstratio
Vadracum AB a eA ad recrangulum ABC, ve AB ad BC: _ ACClatSu
ad rectangulum BCD, ve BC ad C D, id est exdacis, ve AB ad B C; _ FCCFAN-
oulum BCD est ad quadrarum CD, ve BC A CD, HoC en setu d AB da
BC. est ad quadracum BC, ve A B ad BC. Rursum quadracum B C D est
BC, erBO quinq B    B a DC. 810d
crat demonstrandum.
P R O POSTF
M C
u
InCx GI, DF, A C ita sint dinila in II, E, B, veratio DF, ad F F
duplicata sicrarionis OFI ad HII; _ racio AB, ad BC eriplicata
racionis C H ad HI: sincque praeterea OHL, DE, AB, in concinua
anal091a.
Dico e rectangula GHTI, DE. F, ABC, in continua esse analogia.
Demonstratio.
TAEDE DD ad OFE, SIC GF AdR, EE VE PF ad FTY, sIC HTF ad L. igituFrCCTn.
sulum sub DF e K, quadraco GFI, _ rectangulum sub F F _ L, quadrato
i. FTT, oaequalo cA; ergo reccangulum D F K, eA ad rectangulum F P L, ve quadea-
cum CII ad quadracum FII. exlrpochelaurem D E cAtnd P F, induplicata racio-
ne OHI ad FII, hoc cA, DP CAt aX F F, ve quadeacum CFT, ad quadratu HII: ergo
rectangulum (ub D F ex R, cA ad reccangulum (ub B F _E L., VE D F ad F F. FF-
50dK _ L. lince lunc xquales: quia aurem ex hypocheli CFI, DF, AB, _E CX COn-
Aruccione R, GTI, D P, TUNE COnTINUIX; PaTCEOmNeS quatUOE R, CFI, DR, AB CAC
ContinuaS. Jam alias quoque lincas L, FII, F F, B C dico esse continuas; si enim
non linc, hactribus lincis L, TII, F, F, qua ex constructione lune concinux, quarca
continue proporcionalis, quaenis BM, maior vel minor quam BC. habemus ergo
duas concinuarum series K, GFI, D F, AB: L, HTL, F F, B Mt quae incipiant ab equa-
ps. JibuS TerminIS K ex L. quareracio AB adc B MI, cric criplicara rarionis OFTadFTT.
Aequi exhypocheh eriam ratio A B ad B C, crarrriplicata rarionis CFF ad FFI, est
ergO ve AB, ad BC, ica AB ad BM, quod cs absuEdUm. erGOL, HIL, F F, BC CF.
iam sunE CONEInUIX. Itaque ratio composica ex tacionibus C FL ad D E, ec FT L ad L F,
ei hoc cs, racio rectanguli CFTTEad recrangulum DF F, eadem est cum racione com-
Polica ex racionibus DE ad AB, _ BL ad BC, hoc cA cum racione reccanouli
DF Fi, ad recrangulum ABC. rectangula igieur GTII, DEF, ABC Tunc in con-
Finua analogia. Ouoderac demonstrandum.
eR OeOSITIC
E
lc racio A ad B, eadem cumrarione C ad; d llet relaguecam
B quam C, D incerponancur quxvis linex: L quidem incer A, B;
EDEIBINCCIC, .
Dicl
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