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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 3 (shelfmark LH 13, 3)
# Folio 132r, page id 00068200:0265
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probl  de Intervallx Num
contumatio problermatis de intevallis in de differentiis catrium numerorum et (2)
erirum quadratorum, quadratis
jamque ponendo t F, 2r, fiet: - 16 alv- 8al
ceo
v
Ergo 8V+ 8V t + 16a lv = 16alt , 16v + dZ At V + 16 + 2V, 8V FVF. + 877
sed hic comminisci aliquid ut terminum haberimus non ascendentem ad quatratur, difficle
posi explicemus a, quod nolimi; et ideo sic matim
-
2022
in aegratione supra inventa 8V55 = 16 als 165V + 8Y  4v aC
a
oponatur 8Vs 1, 1, egt 8 25 Ly7 sive 4Vs BHZ, et ZB2Vvs
ponendoque V  8 et se r, quod facle est, cum numeri sint hactenus
pro arbitrio, assumti erit Z  seu c - X  2 Y8
ca
e
et de caettro erit 1bals p, 165  + 4Val
vl. 1 r 16419548 a
Restat ultima conditio, ut scilic et differentia quadratorum primi et ultimi sit quadratus
seu re- al t ralx sive 16 r4
ea+ 1d
et pu a, ponndo , fiet
p16r+ +4a
cet
e
23
et ut diffececitu horum quaddatieuu sit quadiatus
b
1
s
8 dlstro, ponendo 4l, fe
a
AY
f.2b
habebimus - 4 a l  srnt F nt et ponendo P, FF= vel = 4a 
ergo efficendum et ut seu b2c
p
ut
sed hic efficere diffiale est ut utrobique fiar. 8+ 
ideoque rectius ab iitro 16r e m l + 4b
tantum ergo efficiendum, ut summa horum duorum posteriorum quadratorum sit quadratus
quod fit ponendo A Y in aequatione 167 +bl
u
n essn
in, et extracta ntrobique vadice habebimus: l r. - 8r - pF 1
Ergo - 8r - p+ e +  
o et redit prior diftrultas
et m l ab  F dbsl. p
fretque  16 Y4  I 8p I- p (2, rectus fortasse- + a  expli cuissemus
ja
util
unde 167  + 4 m  - 8mql + qu  B m (, sive 16Y4 - ta  bnq
et ita
b n - 4m + emg - q, sive t  F+2m - g.
Restat ut ultima conditio efficiendumque u differentia quadratorum primi et secundi
nempe o Brevius forte et simplicius statim ab initio, in aequatione supia
inventa 8Y - 16als ( 165  + 2Z + 4Y al mondo al,  b, A
diemus Restat ultina conutio impleatur, erit quia crraveram ponendo - 4b  pro
b - b + 26. X--
sil ideo rd restituamus. 16r +4b muto
V
et ponendo 4b V, r m  + 2 mG + qu, seu 285 im + 9. fiet , 16   2mq t
. = 16 74 +2mq1 + quC
arbitrio assumto, caetera habentur.
Restat  ultima conditio impleatur,
nec B Ceeeo -  + 2b x + 26Z - x + = 2X Z - 74
1I
n
et pn, pro p poundo +() H X  ye fet ne e 12 - 21yX- 217 Z +X
et cmutatis signis, posito antea neguta affirmatis imajora fiet:
+ 2bX + 2(27
X. 2XV
et pro (V, substitundo 
6
12b X + 2bZ,
et destins destruendis F. V2
vyv, 4
2X LIL
A
-107 + 2l)X +21) ZX -  2 yst
ega ib X  abx
X
habebimus:
5d 70go 4by prgo
eb
() et paano eZ t C4 2X +
pro () sulstituatur S, et pro b substituatur A + S + S cumque 1) sit pure arbitraria
et destactis destruendes ponatur 1 b + d AL, fiet aequatio haec
utanem
2d1b + 4
A 2AIX +2ACy ideoque sau dimidra ipsius , quam X t Z erit et
tum a 12 est quotatus,  gam seprae efeciuus. Euatum et ergo quod quorebamus.