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# Title: [Problema Austriacum Plus Ultra Quadratura Circuli] Problema Avstriacvm Plvs Vltra Qvadratvra Circvli (shelfmark Leibn. Marg. 230, Stück 1)
# Folio 119, page id 1815201908:0176
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ile
at
(C
GE OMEFRICA.
1TG
eruncisieur OF, DR, F. Lcontinue proportionales inratione V ad X: ideoque rario
CTad F I, duplicaca cric rarionis V ad X; guare cum _ rario C P ad F, C, eiusdem
ostendetur, etiam D K esCad F K, vE D F est ad F FT; quare huius quoque prosreso.
do AB, diuidicur in racione V ad X. Ouod erat demonstrandum.
octenta sic esse duplicata, exie C F ad L L, N C F, A B C, C PCMHLAIAO VE CF ad
CB C F C ad F. I, vndererminus aprogressIonIS C E, F C LC. CIt L. lmiM auCuxtu 27G. huims
nis cerminus erit R. Leaque cuin recage piose CL. L.Cn aC. DL, FFT, XC. 2b
Scholion.
ret, divisiones quo que alterne, in eudm semper puncta les R, deberent ineidere; Vt pase.
rum quia in particularibus casibus eiusdem Fheorematis veritas, clarius non raro atque illu
Atrius emicat, cum qura apartuat  casann ooe saltlns de petcistetdao diolets
lium Theorematum demonstrationes proceditur.
Jc quoque voluimns punctum C non incidere in L aut L; ed quod si in alterutrum incide-
consideranti statum Theorematis.
Caeterum qui hanc propositione cum priori contulerit, facile intellie et hanc vniversalem es,
illam vero particularem casum complecti. placuit enim id subinde rum hic, cum alibi factitare,
R O
DS
X V
cecaa quantirades LB, C, D; lirque AB divila in F _ G, ItA VEAF,
Dlic non minor dimidio AB, & FC non minor dimidio F. B; codem
mododiuila sic CD in F de HI, lineque AE, F C; C B, FH proporriona-
les: _ hoc lemper heri pollit.
DicocOTam AB eNC ad eoram CD, vt cA AF ad C F.
Demonstratio.
Tenim non est proporrio AB ad C D aGUaHIS PFOPOLLIOM AF ad C F, exie vel
maior vel Minor: sic primominor. cum ergoponatur AB ad CB ha-
bererationem, quam AF ad CF, habebir A B ad aliquambminorem quam C D b8. Guiniti
nempe ad CK, candem proportionem, quam A F. ad C F:  quoniam ex quancica-
tibus AB, C D, carum que residuis semper non minis dimidio autereur, ICOnEI-
nuecur haec ablario per aliquos cerminos, verbigracia pereres C F, FTP, HO, relin-
quecurrandem OD eminor quam RD; ideoque CO crie maior quam CR: R CI. Decimis
iam ex AP cocidem parres ad mencem propolicionis AR, B. Ci, OI, Eollaneur, exie
exhypothesi AF ad E G, VE CF ad FII, ∞ F C ad CrI, ve LH ad HIO: ideo-
Que permUTandO VE AF. ad C F, SiC F G ad FFI, _ vEF C ad F FI, SiC O Lad FTO
ErGOAVE AP vna anrecedencium, ad C F rnam consequeneium, sic omnes ance- d8. G uinrio
cedences, id est linca AI ad omnes consequences, id est ad lineam CO: sed ve A F
ad CF, sic eA ex constructione AB ad CR; ergo AI, eAt ad C O, VE AB adCR:
quod estabsurdum; ve pacet ex elementis. non estisitur proporcio AB 2d CD mi
nOF PrOPOrtIONC AB ad C F.
Siciam, Siferi pores, proxorrio AB ad CD maior proporrione AL 2d CF: ita-
que aliquac minor ouam AK, habebir ad CD eandem rationem quam AF qq C8. Duinsi.
CF. _ quomam aufertur semper non minus dimidio, post aliquor partes, exempli
Gracia Post EreS AP, F. C, CB ablacas, relinguerur eandem IB minor quam AR.
idcogue AI erie maior qual Le l. Sai Cocdem deean en suancnate CD.
epe Palles CC. LLh AlB, eric exhypotheli, depermurando Ahiade Live LC