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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 162v, page id 00068196:0322
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068196&page=322
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quo radici quadraticae appropiaquant
residuns est e residuns
pagna sit propositus numerus non quadratus 28, radiu quadrati phexime minoris est 4 quae in se ancta dat cb sant 43
2X
in se ductus
adjiciatur fractio sit a4t cujus memta sit cujus num erator sit
A 2sequetiste
resituum numeratar denominata sit
2
si
 1 2.48 v + 2. 45 4 5.3
duplum radicis unitate puctum a +- cujus quadratum est tar
a
24+ 1. qu (2. 44)
quod est mj intans adhuc quam E seu quam I 4. 4 + 5, quia 2. 4. 3. + 5. E, est minus quama
24+1
24 +1
qu (24+1)
eu 2.4 + S est minus quam 1, seu 2. 4 (2. 4+1) + 5 est minus quam 24 +qu (2. 4 +1) seu
=5. est minus quam 2.4. 2. 4) + (2)2. 4 +1 quia
 ex hypothesi numerate est minus quam denominator
2.41
2. 4 +1
qu l2. 4+1)
ilas
semper mau
res fantum eo redit, ut demonstretur duplum radicis partialis unitate auctum
uuam
G
(2. 2. 4) H 2.4
us esse quam  XX +2X+ 1, quia cadit XX+5 inter tanc
lice quadratum berum os proxime majus ita al ratix sit major unitate.
esse residuo, 5.
Hinc sit radix
1691
1681
a + 3
qu. 41 + 21 -.
10. 7 1701
1
20
2. 4 + 1
XX +5
de
A70V 20 1V
et
81
minus
Hujus quadratum subtractum ab
M
id
20381-
2l +1
15
abqucc
doi nrist.
 acedi magis
ccdatur
accedit accedit b +
b+a bta
Datur ca quaeritur C, sit 
d
l= b+
am +Ce, = n, continue accedetur
in +a
2. 4. 4 + 1. 4
sunte a Est ergo serres accedentium b,
2. 4+ 1
seu _ (26C) 4+ b - 1
quaenitur cc quaeritur C, sit cmbb +ac
cc - bL
a = 1, b= 4,  cc = 21
et
9)
cc - bb
b+a
2b + 1
+
in
bac
4. 4 + 5. - ()4. 4-- 1. 4f5 L. 6
bb
= 2bb + ic - bb,. 2bra, = bb fc +a
al + cC
2b+a alta
et notabile est si b sit minus c, hujus serie, terminationem esse
Cnn
42. 4.5
b+ 1
al +cC =m
ab+ cC, fit similiter 1+ a
b+a
m
n
(. 7. 9( 5)
4, 2(+ +5) +4 - 1
ob e ceb +cC ( bbm b - cC (+ bb) ,
ab +cc
m=n seu a m = am + CC unde mn = cc.
qia altimis
erit
f(-b = -ab bb + b9 + CC = cc - ab sinta
2b
b+
amtec
Lm +a
al Le
amicm
eo casu autem au
an+eo
2mm
t ech squ huna
cc -ab + cc - al + cc =am etc. in infinit = C
a ab + aCC + CC 3.ja + 1b +CC A = ab tacc +4CC+ GCC,: ae + db iCC seu b(a +C) + 2acc: (4 a (a16) +CC
sen
+l
a + b
atm
bta, et fiet
aab + 2 acc + bCC,:, aa + ab + CC, sent aa b aa t rahe + cC
b+a
12
a +b
10 = ab +cc 20 = a + b. 11- = 104 10a + 20cC 21 = a + 10
10 = ab + 1eC
12 = 1ta 42CC 1ia + 2ICC 22 = a +11
H
2
a+ b
IL = 10a + EocC
21.
+ arIo
sed 21= a + 10 fiet 12 = (41a + acct 1ocC, ita tertius
ita tertius numeralor it datur ex duobus praecedentibus 10 et I, itaque
t
2 = a +10 22= a + 11 23= a+12
quaeralitur 17 a 13 acc, 14 - acC,
34
itaque non est opus ut denaminatorum serces siugntatim, quaeratur
i 12 = A_  ia + 10cC+ acC 13 =12a + ncC+ ac, n 4L 13a + 12cCd4ac
Ft 22 = Sttacc + fit 14=, 13 = 1aa + 10acCtaa sC + HCC + ACF = 11 (aq + CC) 410acC + Z aCC son 1(aaeC) + ACC (1071)
ed
obatio pisto major est p monet demomi natem lon alta, st 2b
42
d
4+5= cc = b.
Eneit m
un2
es