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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 340v, page id 00068196:0676
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ses A irrationalit partem, et  ea v fit 1on1o a 303I8 = 1. 70. 190. 240 +  2. 20b0
propterb V fit 10. 140 + 50. 340 = - 2. 38140. 343. + 2. 20 B0) po d, (P fit n. 140 +31. 390 2 - 2. 31. 1403 40 + 2. 2. b0
in lesimali prute prosic. V fit 1e 1o 130. 70 = - 2 .  190. 340 + bB. 2 1013 4 At V fit 1, 43+ BI. 3 45. 3 -2. 1 H CC Mo. sH +  bo2l. 1r 31 + ba
proper v fet - 2. 704. 140. 3 40. + 2. 20. 71 b0 - 270 31 140. 340 + 2 2t 10i be (, 0. 217
ex aq tetr fit  + 2.10 31 . . . . ( era + =. 1 3 50. . . .
Ex 1013 fit (.  2. 10. 3050 - 2. 78, 190.340 =    20 b0 ror3a - 2. 20.ib
Ex 2 et 4 fit
x 6 et qe lit
190370
d5
2. 11. 31. 50 - 2.71, 140. 340 = 64 + 2169, 1131 - 2. 21. b0
20.50. 10130 b0 = 20. 70. b0 = 2. 10. 30 5050 2. 70. 70
20. 50. b 10+30 bo . 2 20. 70. bo 17 21. 50. 114 31 bo = 2. 21. 71b0
f. 10. 30.50. 50 - 7. 70.70
C fei
7. 11. 31.50, 50- f. 71. 71
e
 20 ((B50. B = 2. 20. 71 - 2. 21. 30b, b0
10) Y 20. 31
140340=
per aeq. 5
21.30
 2. 10. 31.50.50 + 2. 11. 30.50.50 - 4. 70. 71
Htabemus ergo tres valores lipsius 140. 340, adeoque dua aequationes, a quibus abest absunt
140, et  quibus si dertia adhuc accedat, ab his independens, habetimus quae sitam
sufficientes aequationes sine 140, et 340. Et quia ope aequ. 1 et 2 invenimus valorem
ipsarum 1 140 et 340  pagina praecedente. jamque per  s
83 et 4 conjunctas cum 1 et2 ad huc novam aequationem obtinuimus et novam stidem
per aeq. 5. Superest jam ut novam adhuc obtineamus per aeg9. 3 et 4
inter se conjunctas
 +byl C rhc
Cx + 
XyaB) p
e compendum pro 12, 307 ag. 3. et 4. sembamus ax + by = 3 (e rmy =n
 pro 140, 34
talX+ bly = et
alXX- amyL N’a
argep+ bryy = er+ n5 su bt an
l5.. +45..
ar+lS = br + 15 d a - br = m - 15
fit r: 5 = M = l, :, a - b
it yy = 10- a1
 fit tarXX + bY yy + ar xy = er +ns + par
ar + l5
p sun
y etva.
Hoc suopone nocanda
(TY = mC -b
b
seu a Xx = gy + 2 xy = Cr + n S+ ar+ (sBpp
 sit s-1 et Y = ma l 5
et fici XX + 9y + 2xy =
ry -
seo
X = S me-(C + an - bn + rp an. Bl Hm C le + a - bu - 2p am bl
y = Ime - lC + an -bn + 2p amabl = Lme. le an -b 2 - 2 p an bl
et X jignifiell 140. et y fignificet 340. patel
sed quia in C3 et n rursus labet 140.340 et fit 21= 20, 10+ 30. 10. 31F11. 50 = 10. 3a. 1+31 3: 10. 20, 10430- Quos duos valores ptru 21 exegg. 11. et 13.
itam el fit xy = LCn amcbl - m - n ab - 40340
i vel 8, vel 9, ve
hic iterum haberi 140 vel 340
=2. 9 70. 1131 50 Xy ( f 10.70, 2, F 3, b0
Un rkab - Cn anebt
 l  
na - 31 q F ffser 10. 540 =
 inubrru o sepliolle. nve t 2
initio hujus paginae,
_ NGC ep. fit C3 = - 2, 10. 30 50Xy F bo. 2 10 430
X=  se  nt) Fugo vdia
ser
seu XX+ ryfrpus = em - cl f an -bn 4 a nc a l + a b1  s
sit eX + 9y + 2Xy = hq et xx tyy -2xy= ht
fet x = 1h9+ iht et y = thq- sht. Ergo
finc si (1.). et ((2
bltam10. 31. v. )
n’a  etfiest
bI - am
XL9 = XX +yy + Exy = em -c + an-bn 4 am- bC 2p
am = bl
n3 = - 2. 11. 31. 50xy + b0. 2t. 11431 C71 
Ny- B sC l.na fibpanXu, = + 2 l. 7 1 1090. 5CXy  - 1.31. 20. 101 5 a b0
bt am
C. tuil t n. a. b + a nil  l. a. tB tip
a. m- l + l. a- b
am= al) + C+ ad - bl
Sam- bl
tam- bl
5.7
am - bl
X. ideo 
non sufficit, sed pretius omissa
extractione scribendum
Ax = 1n umebt- sim - nab 
av. 34
(2) am - 6l~ Bet
et hoc explicato, tandem nova extra.
chore habebitur X G, pova
recg demum erit aequatio ab B et
Vage liberata qui novus
valor cetsi irrationalis, si cum
aliqui ex ipsis 8, 9, 10, conjunga
tur, habebitur aequatio nova.
Nempe ex valore 10 et 8 fil aeg.
prima, ex valore 10 et y aequ. 2da
ex valore et nova inationali
et valore 8 ag. 3tia. Et per
has tres aequades liberas ab, 40
et 340, psa habet praestabitur
oparatio ratabsolvetur.
in ag. XXyy seu 140. 340 = 
explicita pro xxjus potent substitur valqu
 psius ortus exvaloribus desu invicem ductis
ita habebitur x9 rationaliter nova aq
quae esse qui eut varot 1) qui junjendus ipsi
 o dite t a e ete a ea tut lee
Hab