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# Title: [Problema Austriacum Plus Ultra Quadratura Circuli] Problema Avstriacvm Plvs Vltra Qvadratvra Circvli (shelfmark Leibn. Marg. 230, Stück 1)
# Folio 112, page id 1815201908:0169
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E. SSTO N F S
289. tninc.
Gressione BC, C E, F C, _c. e cundem cerminum habet: Acqus progressionis qua-
druxix BC, CF, de c. terminos FI, Secaca C B in ratione rnius 2d duo, ergo CMaH
cerminus propressionis AC, C B secaC C B, in racione rnius ad duo. Juare C’Hl di-
midia est ipsius FIB, _ C B selquialcera ITB: ideogue A B cpIa IpluS HB; ac
aenigue A FT dupla HIB; ergo cerminus propressionis AC, CP, el. UCat AB, in
Lachonernius ad duo. vlcerius cum prooressiones AC, C B, dC. BD, 3 PB C.
_c ad rocam seriem progressionis BD, D F, _c. b ve AC ad BD: QUarC CHM AC
dupla Ac DB, cric quoque series propressionis AC, C B, ecc. id CH A HL. AdPIa
seriei BD, DF, XC. Aequi AEI iam ostendimus criam duplam ese LLB, C18
AFI, ad seriem prooressionis BD, D F, candem habecracionem quan a LIB. Vu
de EIB aequalis estleriei B D, DF: vnde prooressionis BD, DF. setres l
cur eriam in puncto TI. Juare per eadem procedens puncta, cum alcerna illa bile-
etio Constiruar veramque proxressionem, illius quoque cerminus erir punouu ll.
qsuius. eiuldem sincrationis, nempe quadruplae, exie tora series prosressionis AC, C E, E C.
quo dividicur AB in racionernius ad duo: quod erac demonstrandum.
Corollarium.
X hoc Theorematereperietur areuscxilectio, s independencer L rrilectione, al
Lcernx illius propressionis cerminus inveniarur. Cum enim Theoremarniuersale
SIC, ex in quavis magnicudine demonstracio allaca valeac, s in arcu daco A B, Similis
ctio repCrICEUT.
alcerna Aac bilectin cerminus quoque progressionis alcernae FI, abseindet cerciam ar-
cus parrem BFI: proincese lepelto alla ba alclorermino, arcus eriam dari crile.
Aca sic magnicudo AC vrcunque secta in B: Deinde HaE ve A C ac
BC, siC BC ad BD, d BD ad E D, _ E D ad L F, _ L. F ad HIF;
Dico verim que consticucum iri duas progreltiones limiles, magnirudi-
num AB, BL, LLL aC. _CO, DF, FC, ac. continue proporrionalium
in ratione duplicata proporrionis AC ad BC.
Demonstratio
Voniam ex dacis A CB C, BD, ED, E F. XC. III Cmnux proporrionales.
er CMeCHCACAB, CD, BF, DF, FH, FC. XC. CCaMIMCOMMMUA
Thalogin, et quidem eo ordine reprima, cercia, quinca, seprima, e sic deinceps (in-
cermisso semper numero medio) constimant seriem A; Secundo vero, quarta, (exta
Octaua, _ Sic deinceps (semper omisto numero medio) seriem C, conticione. igiturve
AB ad BF. prima adcerciam, (c. CD est ad D F, secunda ad quarram; d sic dein-
ceps; adeoquerationes AB ad BE, _ BB 2d E FL, exc. limiles erunc racionibus
CD 24
atque sc alterna divilio semper Har;
Vi   i  .   1                   u        ue    s   n     e e n   id