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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 3 (shelfmark LH 13, 3)
# Folio 132v, page id 00068200:0266
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Sroblema: 
Invenire tres numeros, quorum differentiae quadrati;
et differentia quaoratorum etiam quadrat;
100
100
Calculus hic est, sit primus numerus a, erit secundus al = X, ita ut differentia eorum
X sit quadratus; et tertius erit al - x - Z, ita ut differentia secundi et tertii t
sit etiam quadratus. Jam etiam differentia primi, et tertii, nempe X + Z1, debet esse
quadratus.  Quod ut fiat ponemus  + Z et y + 2CZ + C f 1Z  Xt 2CX+ C
et erit Z  X +Z~  Z + 2C7+ C2
 seu XX-C p Z. smatis coet xpro utia
Jam videamus an differentiae quadratorum quoque, sit quadrati possint reddis,
ergo Zal X A 7dl X dC
d vq2al + vV
21
1ergo X 
ral7 v
2V
Differentia inter quadratum termini secundi et tertii
(N
neanempe
u
Salt d
4
--aju
∞ - 6:
-b4 + 9b =36
2go n4
a l - 24lX+ X - a l + 2al X  + 2alZ = X - 2X L
=2 f
 sive 2alZ = = X v7 b- Z4 equalis ciundam quadratoAZ W  divisisque omnibus
perfZ habelins positoque W  Z + 2fX +f si scilicet differentia inter Z. et f. minor
est, minore ex ipsis; habebimus. 2al -2XV = Z ⊓ - Z +2fZ- fa
Z e 2al + f2
1 +
-
v
C unde ralc +fC 1 Xf = fc
c
 sive
ponendoque, C. B X + g. fet nalX + f + 2alg + f X + fg F, XY - fX = 2Xgf. g7
sive XY = fX, se destuentibus, fiet X  = 2 al g = gr f ponendo 29f p 2al + f
2 al + f - 29f
Restat ultima conditio implenda, ut scilicet differentia inter quadratum primi et
edi tertii, sit quadratus. Ea differentia est.
at- a t +2a lX + 2a(Z - X 4 - 2Xiy v 94.
aequalis quadrato cuidam - Ita rlh
vel 2+42
cuus quadrati, radix lh - X - q, ponendo di eam esse minorem quam X +
afa
im  etg
destructis ergo destruendis, fiet +nal X + 2al Z  - h + El hX + 2lh Z cumque
Ih sit pure arbitraria, ponatur r la + Im, reformataque aequatione, habebimus
vV
la + l’am + lm r. 2lm X + 2lm 
et pro in sumendo D2=
poterit utrobique radix extradi, ac fiet la + LN py X n SX +Z Nam
y
L +X quadratum esse effecimus jam tum supra,
Unde habemus problema
solutum indefinite, ita it trium numerorum quae sive la
fierque la +An
et Z e 2al Ff
25
C
2al + f2 2gf
Jam g. et f possut sumi pro artitiio ergo sunt indeterminatae
ergo quem admodum et a, et n
Utergo a, assumi possit pre arbitrio, tantum efficiendum est, ut quadratum ex
Unde
alicuu ex literis, a n. g. f. evanescat: mimirum aequatio haec est
la +an rien al + nfr seralgn - ngf +ng:)
zal + f 29f
- 29fn
2al + f- 29f
2al + f 2- 29f.
Epofta 4 e, HX al 4 XsE, in Ergo. ifla + 2fnp e 2n ab e f a. diris
aX
 PH s, er f ni  ex posteriore egntione D, eor prn D dabit F p 245 tatu
cem
il +2
al 2f 2gse 1g 5., ten ras at 5 4 188, 14 X ral r 87 teo 49 ft 
eaque reducta a lf +faltu 3 la + pn - 495 al - 29f n p ad l’nrrabif t entr
+ nfa analgf nf9, - 4g fn
Sumtis d, g. f. pro arbitrio, et NX e
Sed brevus satis faciemus,
= 29f
+ nC