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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2a
# Folio 55r, page id 00068197:0109
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D10g+  1DE
Quaeritur ergo fdx S(Xx tab, XF + ac) fit x = a,
fiet 
Xx + ab =
et xxtab, in, XxtaC =
aa 0 taa D + 2aa Q  P D
abO D
ego
VO+
aa0Q ga DP + 2aa 4 1C
et XX + aC = - acO
 aes +ons (4  OV
42b5Det Da bordab
BC ( 1E+ ac Cp  aabC
+4a 09
44+962
24b500
10
hc On
Htius aqatieea Fractionis Numerator ita explicandus estnt   inde radix quadrata
extrahi possit. Et eo pro latere ejas assumatur aam dBQ +  aa 2 1DE, cujus
quadratum am+ 
s sid hic obstaculum video. pars rationalis identificanda est rationali, et cum unum latus dividi possit per
quadratum erib.aum  + a 44 DC
ben aper ubique praeterquam in nabi CODD, siquitur etiam OC)ND  latenter, quidem di divisibile esse
per Q, unde cum  D e C, et 3 ot _ sint simplices sequitur alterutram horum esse priorum esse in
constante ratione ad tertiam contra intentum
Ergo _ mutata nihil Hypothesi, fiat X =
aa 02 +na DE +2aa G) 1OD
fis huet tabee ab extabtih
Xx +ab =
OD +tiq +2t 1CD
104+ 1+
1OD+t
et XXtac =
O + hh + 2G16
Fet i Dohirhtalu si ducantur hi duo vabres invicem, erit, denominatorum quidem productum
quadratus sed Numeratorum productum aequandum est quadrato, quod sit acaer
a Latere 4 AC + Z +DE, cujus quadratum est 24 08 + Y )Q + 2244109. si am
cdefficient partis rationalis identifice in hoc quadrato identificetur parti o irrationali in producto
ex numeratoribus, pabebit in aequatione identilicatoria munia membra di, ex ceptii aliquibus in quibus est t. Unde
a htanenu ad E tn ils 2  pi ration conparse edita ehgrus tratiore  tartum
tan sutatiam omnia erunt divisibolta per 4 praeter 1BDD, ergo etiam B ntD reculetur ad _ quod non licet.
Ergo optis est  Hyptothesi paulo manis composita etfiat X=
froD + freet sive tat
10D + sC + sD + 2
(50) F + 1750101
aaitiue
ab67
ab 2j2
4 2a55 10n
prinum +
ilam ra
 46)
Et curabimus comnes  tres irrationaes in toto hoc calculo
124
ic id enim modo non aliunde impediatur
calculum reddet breviorem
40
72.4 S.
pro compendio st tab =4 Hstac = ah si iil. H
et ipssus raxtab Numerator ert a, 9e
dtit
h
quibus Nuneratoribus inirem ductis siet Taa, qugus ex) t  h 2O) 4490GB
e
644
agtt 
ani
B+
d
ab 22
at. .
4m e0
aticum
igumeor
no
se
t.
Ha productum, ut ostendatur extrahibile quadratice, fingatur latus OD +E sE F.r ta
cujus quudrulum est a, 50)+ 98B +_B0Y sii + 2a 5  +  2909 tOn
Hoc identificandum est productu, et primum pars irrationalis irrationali.. Et notandum est in parte iratinali quad
quadrati nullum  non comparere membra ubi  dD et 1OC et sBE mulsiplicentur per coiucidentia
+948+ 
seu eadem. Ergo in producto apgrequlum omnium horum membrorum fiet aequale nihilo
Et fiet g6)
F
ED d. d ggt 6)4
+ adi2mZ C +Bgh CDG =0
math. L
n in a0 
 d)
eabs
siet Xx + ab =
+2b810n
+2abh10)
2 a M 27
manifeste  alterum
aaaBq
accb tach
72 aa 207
2act
nifestiu
C dant 
Aaa idem quodeminq. nisi
Harum fractionum denominatorus invicem ducti, cumincidant, pnotacum, u
Aatac  pra b ponlturc
b = m ftC = n
+2a, 9On Et ipsius xxtac Aumerabient a, ROY
+2. 61O)
1000
22c)
+nOr
traeson
nZ10)
4to + 8gli CB fiOY
nGYC
etox
a hae
C1o
Quun ht e
stqu