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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 3 (shelfmark LH 13, 3)
# Folio 127r, page id 00068200:0255
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Ires Numeros invenire quorum diffecenbia sit quadratus
Tres Numeros invenire ita ut duorum quorumlibet differentiae Cail F Z + X  M
summaeque sint quadrati.
Numeeus pumus. Vat.
serid
Numerus dy
Num F.
Num. 2
Num 3.
at 
al-
al - X Z
4) Dift. Numg r. et. 2. X2
) diffra Numeri 2 et 3. 72
(6) Diffra ctumeri 1. et 3.- X (2) ⊓ C) . Z (Z
(7) Ergo
s X2 = C  bC - c
+) 2C
(0) summa Numeri et 2, 2al = X,  2
(9) Ergo.
(10) Numerus 2.
(11) Num: 2.
(12) Num. 3.
a e d + X.
d 1X
1 =x
a X 
F2
Ergo Z 
d X 2292
(14) Summa Num. 1. et 3. (4) - Z r(d) ( 2dg + g
(15) Ergo
al -XV
N tal - X, 2- X a + 1.
1 -at e aleX Z+ K
al - X Z ∞ al- d
in his 4 aequationibus continentur
omnia Responsa possibilia ad quaestionem
quoniam nondum ulla facta est deter:
minatio, atque inde eruere deteret, qui integros
minimos possibiles quaereret
-2641
puc en h, fet  1
2 seu vidinum. 15, 2   9
poeps,to 
(14) summa Num 2C, (1). X Z  (  )dg 1 g2
15) Ergo
(+) 29
a
41
seu f 4 gr vet num. 16.
seu 5+(4
1610+
41+ aequari quadrato
+ 1- hef 
4(4  I
a1) N E
=14 157
Z fiet 5af 25 h )vcl 1618
16lo 4(4
siom 16h + 5 Na
fiet n aequali S
4 N   vel 572
IT.
ponatu 
5+Y , +
d F. Z- g. sive per 3. a V,   et poundo C1 ED. fiet dpj 4 14 = 9 et p2   
(6) summa Num. 2. et 3. d.- x2 yk
++
(1)2 fal. 3,, 
Ergo (4) ponefti.
vel (d (4) 2dg + 9- X S1 (d )idg +9 3)  sg + f 
vel -h 1
sver
et 5 
- h et. vd. pumit.
() A
)
( NF ) afC+D9 sg + f2. d 1 g CCFD5., dant quadratum.
Ponamus L sq fire divisis omnibus per f fiet:
L 1
a a c) dt tud 1g d Wis
h fa g. e C+ 29(g (119+ f. Pdans l r 4
Ergo f habetur pure cempe
f,29
+2647985, t 4
onans h se
Ponamus 1+(+ s 13f+ f 70.
eet
(+) 140
D +S
() et
f + vs + H g r sed hac divie.
f. +Cd H0.
Sed ista duisio aequationum per alias Aequationes non est, nisi raro et per accidens apta
instituto praesenti cum enim duos praebeat ipsius f vigs valores hinc eos conferendo
(nom enim hc de potest ipsa f. eaditur. hud minime ut jam vided, tantum enim
duo habentur iptius F valores, seu rudices aequationis, id est utroque modo satisfit, apta
+ vs + W ig 9 f 4 v co epo Ergo vC -Co e _ g2 Ergors  
ergo inisi fallor inftituto nostro haec divisio censenda est:
Acof- cos
F - co + c 
su + cof
pone q E co, fiet: vas- co - W  cor
et cop dest2 Ergo fp ut
Wx + (3
Eve
WX + 13
Nac mihrodo divitendi utcadum, quando non licet Quobus valoritus in locum ipsarum f. et g. substitutis videbis
is e. vo
sue pre ut g. scontneas  vel contra, quia
n continet i
iu numeris rationalibus, v. L, W. X. pro arbitrio sumtis
satisfi eri aequation. Hime jum quaelibet aequatio quadratica
affecta, nullo negotio in numeris explicari potest, facilius
quam si eam volut spemus apta re ad extrahendam radicem.
Si Radicem inde voluissemus extrahere sic fuisset procedendum;
Quando ergo N quadratum est, ut in nostro su aut facile reidi potest caeberique
V S + vs + W gr 10
V hiet    gr
 +2F4 P
apta sunt, in nostro casu, etiam extractio commode institui potest
ne sri a ctg i 9  C 9et  s   sp. 2. cen moo E si n. iga
2C2
3400. X (