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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2a
# Folio 54r, page id 00068197:0107
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# After the first blank line: the recognised lines in reading order, one per line.

Xx tab, XX + ac  qu  aM8 +  YOp
02970
etatue +eroi
9. proponitur jam
fLBD + 91 08 +h soc faal
nYCG
Quaritur (, ex (XX F ab, XX + a6)
sit x = C. VOD + f +  + gY v9 + als
704
bh. 41
mEr
f 10D + g. 194
= at aDb F 4 - a) +ch
aa degabdE t
a07 1447())
se
ende srgo odd visib per a v
sirt Xx t ab =+refifg er traletor
Sit X
abbe
racs
Na
aa)D+ cac fet
murrie
e ura manet irrolio
destuantur irratipale
fiat fg =
Xx + ab aa9 BG +rae ab tr 10E
aurjamse
ultima medietas
nast et t108
abte C
abaa tu2
altim
seeD
aatut renos 
Eb bi
e10e + aE
Eodem modo habebitur Xx + ac, tantum pro b ponendo c. ducantur jam iavicem Xxtab, et xxtar
eqquidem hoc ducto non potest tolli irrationalitas, neque enim aafu neisi cas  ee c b hac
 neque enim potest esse fg + eb  sig ec in hoc producto tolletur irrationalitas
si sit sig fgt ec = sg lentemusque primum an non in producto possit tolli irra-
tionalitas. sit. Valor ipsius Xx tab, compendio sit 5 + 6108. Et valoripsuis Xxrac
debet esse A n, 2 = n5 SdQ, ut pruductum fieri possit  mn 22 L mn 8508
Ergo 4 = aaOD + etc,: am = aaff0D trete,: an
et fiet aff (mun) OD + agg (m-n) DQ +  ee (Bm - c2) 1q + aalbm - cn) hZ = 0
Rursus ob 5 = 2f9)  fgD + bet,: m = - 59D - bet, :n, siet
bi + m) 5BF
(bn + Cm e
refg 12+ m) FgD +  (on + Cm) et = 0 Et habetur F =
reperitur, fit af(m-n) 9D + agu (m. n) D e + ve (bm -Cn) DQ + aa qu ( n) 5t 4) 97) -0
qubn + Cm) Ce
Haec aequatio cum sit virtualiter identica, necesset est, ut O4, nisi in  destuatur sit
divisibilis per 5N. Sed s non potest destui, nam si fiat bm -en = o evaresceret etiam
9) et fieret aff1D + 9gD =0 unde fieret ff + ggG =0 unde O et e haberent
inter se relationem constantem et totum esset reductum in X ad unam irrationalem 1D,, et
(00 foret rationalis. sed ita nec opus fuisset aequatione t qua _ in venimus valorem ipsuis t
quodsi aequatio as (m- n) O + ete sit dirisibilis per C e sit divisibilis per D necesse est ut
alterutra C vet ex habeat rationem constantem ad . Esto e quae habeat rationem constantem
ad D et 19O ert rationalis, et omnia erunt seducta tad unicam irrationalem 10E sed
esto superest  qualio  aama mnaa 44- mado = apeZ, ubi notandum est 
esse s esse in ratione constanti ad D , su 6d - (9: a) XE∞D. quanquam istud
59 f e SsC +gg D non servit nisi imaginarie cum O et  habent eadem sigu
rbnque praestat ut productum ex. X+ 4 ab et Xx + ac non sit rationale, et tamen
penducal quadratum sit  58+ 9t pGR +q) in rese t stB videamus
an haec unicenducta possint producers quadrat, essi non snt inricem in rectione constante
sica p pEE +5GO SE (5 1G) SC D s hoc non procedet, s etas est ulta s ponamus radicen
licebo faedre esse ejusdem forma: cujus producentes. Tit ergo radio + O +5+FD, et
a L  et frit 540 + Z jC +  253Y +Ct pos si com–
51( + YYC, fin quadratum s NO + 22 + 25V 4YOE
paretur producto ex valoribus ipsor Xx + ab, et Xx+ac fint 2N 4 (27gaa Dtrebrat) inotri
et quidtu reductum est ad D, erit et N, et  reductum ad D et fint at boE tet in aqueti et, Xst C ++7)
se seu reducetur ad 5 x BGC) et B  B quod ditet se aejele facto ex ductis in vice ( +9B in m D)
et dideslibitur etqe rmtat (3) + B producta ex D vils Cdt n( g eD in C q.D quod videtur fieri
posse summum terminum distruendo in hoc producto, itaque habentur si et  per D et B
Nunc ergo Dr S YY (F a0 + tB cut D D 5 pe Et srit C) (DD Futh t.. DZ) + (. DD +uiti F.. DG)
=fato ap  (abE) ex  ( (a blh + etc) (aeth ret) + De (m. BB +. anti  + n.. DG) irntiona
ubi cum unum latus aequationis sit uno gradu altius altero, poteritt aequal si summus altioris
imo debet fieri 3 10p + Z1D, et fiet ct
potis insi nter
identificari si sum nsatintu
it oo se  oimast u 
at olos  ui i  ali  dtiu la i p 
sit
ata
Aat 2aafg ∞ 2aa
es sa raa eotue raaebu=
 indat ~ rm 7
aa ()V
aa DeD
20a 6831
tto
X = a
10D+9
at 12101
sita