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# Title: [Problema Austriacum Plus Ultra Quadratura Circuli] Problema Avstriacvm Plvs Vltra Qvadratvra Circvli (shelfmark Leibn. Marg. 230, Stück 1)
# Folio 99, page id 1815201908:0156
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1d
B.
Term
ci
Det
time
FUM.
TanE
educt
i
p10.
nea
ale.
con
e
po
GEOME TRIC A.
Hypoche, t c quam AB; ac proinde cha BEV. Cu XLML, BH PFOPOTTIO-
nalessincipsis AB, BC; Ccisicur concurtus in R; eric ve AB ad BC, SC MA
2d NB; sed ve M A ad NB, sC AH al BK, ergo AB cA ad BC, VE AK ad
BK. Vude AKcori seriei a qualis cst. Factum igirur est quod perebatur.
LO
DSTTIO
L. L2 a L21.
lemagnieudo AK, Peries eoca rarionis AB ad B C continUAEX in
infinieum.
(AB ad BR, VEBC 2d CK.
AB ad AK, ve BC ad BK.
DicocHc L AK, BR, CK, ec. continue proporcionales.
1AB ad BC, VE BK ad C R.
(AB ad BC, Ve AK ad BI.
Demonstratio.
IX A B abCcindatur NB, aqualis B C: A ergonOn est A B ad BK, ve BC 2d
LCK, neque permutando AB eric ad BC, id est BNY, ve BK ad C Ri ergon-
que dinidendo AN, eric ad N B, veBC ad CR. FiaE ixiEUE vE AN ad NB, SIC
BC ad aliam magnicudinem CM, maiorem vel minorem quam C R. Teaque com-
Ponendo, AB eric ad NB, hoc CA BC, VE BM ad CM: d permurando AB ad
RM, DE BC ad CMI. Ouare corius serieibtacionis AB ad BC, TermINUS CFIE M;
sine AM aequalis cric lerici racionis AB ad BC, quod est absurdum, cum A R n 2-
ior velminor quam AMI, sic ex hypochelaqualis lerici daca. non cricisitur alia ra-
rio AB ad BK aracione BC. ad B K, ergo cadem. quod erat demonstrandum.
Reliquas aurem asfertionis parres ex prima deducemus. Cum onim iam demon.
Aratum sic ex hypochesi cheoremaris, sequi AB cNC ad D K VE BC ad C R, compo-
HCndOCTIE AR ad BK, VE BK ad CK: quod erat secundum.
EC quoniam AR est ad BR, VE BR ad C R. igitur per convertionem racionis,
AK estad AB, VE B R 2d BC; de inverrendo AB ad AR, VE BC ad BR. quod craC
cercium. rurlum quoniam AB est ad BK, veB C ad CK, erir permutando AB ad
BC, VE BR ad CK. quod erac quarcum. denique quoniam ostensum est AB esse ad
AR, VEBC ad BK, etiam permurando AB est ad BC, VE AH ad B K: quae omnia.
eranc demonstranda.
79. huius
Corollarium
X quinea assertionis parte hoc cheorema deducieur: data hc series maonicudi-
Vicoemre vna ancecedencium, nempe AB, ad unam consequencium BC. (10
Sumes, hoc est intinicas ancecedences, siuc AK, ad omnes liue intinicas consequem
SCd, lvC BK.
num continue proporcionalium AB, BC, CD, eC. Sine termino ContinUAFA.
SITIO
EX XIIB
IE AK magnicudo producta ex ratione AB ad BC, IH MLIMMTCHM
cinuata; primi aucem e sccundi termini differencia (IC AMI;
Dicoprimo; difterencium AM, um, e CKtoram
seriem,