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# Title: [Problema Austriacum Plus Ultra Quadratura Circuli] Problema Avstriacvm Plvs Vltra Qvadratvra Circvli (shelfmark Leibn. Marg. 230, Stück 1)
# Folio 143, page id 1815201908:0200
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erie
e
DAE.
2
GE OMETRIC A.
Demonstratio.
ErieS quadraTOrUM MIK, 1X - 2 139. hM-
quatur rectangulo O AB. 8x ins.
Teries quadeatorum NR, rC-
cangulo PFC aequalis cA.
Acqui racio rectangulorum
OAB, PF C exlacerum AB,
FO, _ AO, FP, rationibus
componitur, ergo criam lcrie.
TUM NIK, NR, ex iildem ra-
cionibus proporcio componi-
cur. Ouod crac demonstran-
dum. Sed hoc Theorema vni-
vertale reddamUS.
PROPOSTTIO CZLIV.
Adem manence Rgu-
Lsta, daca linc series bi-
na planorum similium,
quarum longicudines lunE
AK, FR: lIFAUECM AR
amila in C, in ratione AB ad BC, e F R in P, in racione FC, ad G FY.
Dicoleriem plariorum similium MI, ad feriem planorum limilium
AFR, habere rationem compolicam exracionibus AB ad FG, e AO
ad FP.
Demonstratio.
pex ijdembalibus hanebinae quadratorum leries, de lineis CD, HI, Xquales
fiane aB, EC: deinde ve Aa, ad AB, et BE, 9d F C Hac quadearum A B, ad AB
quadraru, ee quadratum F.C ad quadratum P. B. ieaque quadracum super ABx qua bi
cur b seriei quadratorum MR: planum vero super AB simileplanis A S BT, EiV.
GX, ec. aequabicur seriei planorum S R: Cimilicer ab alcera parre quadracum lu
per FB Cerici quadratorum NVR, e planum simile super cadem FiB, planorum (c-
riei V R aequalia CrunE. Ieaque series planOrII SK est ad seriem planorum V R,
replanum super AB, ad planum FB: Irem series quadeatrum MK est ad seriem
quadracorum N R, ve quadrarun super AB, 2d quadeacum super F. D. Aequi pla-
num AB est ad planum F B, vequadracum A B. 2d quadracum HD: (cum enim cam
quadraca, quam plana ex constructione sine similiq veraque sunt in duplicata ration
nis AB ad F8:) ergoseries planorum S R cM ad seriem planorum V R, ve scries
quadraEOrum AT K 2d Teriem quadracorum N R. Acqui per precedent. leries qua-
dracorum MX ad seriem quadracorum NR, rarionem habet compolicam, exracio-
tBuS AB ad F CI, eE AO ad F P, ergo _ planorum series S K ad seriem planorum
VR, racionem habec ex ijdem racionibus compolicam: quod eruc demonstrandum.
PiOPO5II1O CLL. V.
3130. Lu-
Laca tulaulatotula alao Uliul ei lcul lto Lactin
aqualibus quadacs elpientes: sicque linca S Cericirarionis primae
AB ad cerriam C O, Gue linea T, Ccricirationis DP. primae ad Cer-
Dicoleres quadratorum cam proportionem habere quam linea, 5, I.
Ddemon-
ctam Ll aqualis.