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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 211r, page id 00068196:0417
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X + sp X + aqt taa X + as xx P a t X  av = 0
fiat X + pX + agX t aar X + aSXX = ee x + aa fx + a’g XX + a FX + av
+cex + aa fx + agXX
id jam agitur, ut  utraque pars sit extratibilis quadratice, seu ut quadrati sint
tam XY + pX + agXX + aarX + cs, quam eex + aafx t aegXX + a EX + av
identificetur eex + aa sx etc quadrato e re ab ext D X + aac quod
+ ee
aaf +ag
+ aa dd XX + 2 acdX + acc  eex + aaf X + agxx + a tx + av
est ee X + 2 a de X + 2aa cC X X
fiet C = Iav et d = at: tav et proinde habentur, c et b
et f= 28 C: a
sed video, si jam ad alteram accedamns
cum quadrato aequationem non esse sufficientia dta arbitrarias.
et g- 2a 2 ce + dd,: a
qu (Exh+ ad xx taaexta )= CexacdX aacex r x
An sic (X + X + agX t aar X + a sXX + a tx + a v = 0
(melius.
Ergo
+aaddx aedex+
IEX PaeS_ta
4CX
u Eai radxe aa et 
siet exX ACx + aepXE+ aaqX + aCYX ta C
ab X+ abpx t aabqu + abrx absxx + ab1 x + abs.
CCx + 2acdX +. 29a CeX + 2a cfx
+a DX +radeX + 2a dxX
ta ee XX Y 2efX + a ff
= CC X +2acd X + et
dat
extracdXt2aa cex
Hoc latus sinstrum aequandum erit quadrato a  yX + DXX +. H X +
Habemus com identificatorias septem, artatrarias autem quantitates novem
nempe c, d, e f; ), d, H,  et praeterea b, quo multiplicavimus
aequation
datam aequationem V = N(ab YCC)
B = abp + 2scd,: 2  s (ab + CC)
p = Abs+ S
H = abt +2ef,: 2 165 + s)
supersunt tres aequatiores, ubi hi valores sunt substituendu
quadratum a Vp + ADXX + aa HX + ae est DyX + 2a]BX + 2 a a V HX + 2ay GX
+ aa bD XF + 2 BDH  + 2at BGXX
sit jam _ ex ipsis X, aabe bg. +rce + dd = VB Ab L + Zs) s a 4 cd
1bs +
+s
ex ipsis XX sit bs. + 2df. + ee = (lp + 2cd)(,d ab +ec
+ qu (bt + 205)
4, b5 +fs
Si fierent A = 0, et H =0 mullum contraheretur calculus
5)
et fieret bp + 2cd =0 et bt +20f20 et bf + 2df = bg + 2ce + dd =0 et bs+2df + ee-
quibus quinque s aequat ombus
 et superest br + 2Cf + 2de = 2 +(ab+ cc, bs+ )
22gcd. p +28 - - 29 cf: + + 2ce + dd =0
2ot
=29ef - 2tce = tdd, fit
= pee
2scd = 2Bd5
sed rmrsus d = X Ps  c.
pf.
et = et et = pe
in irrationali sunt b, c, f. Et
b = -2cd, : p = 2 ef: t, fit cd: ef = p: t
solvendis inveniendae essent literae c, d, e, f, b
extra irrationalem sunt d et e
= 25c0: p +2d5 + ee =0
2sC - 2p f et fit de=
f - 2tC = pe
+d
0+ - 29f -2tc
seu C = 2f, sc - ps,: Ct et d = 2C, 9f - tC,: P
+
2SC -2p5 et fiet
20
sc pf, et et
p.
g5- tc
=p
Habetur ergo F per aequationem
A Sc- ps
quarti gradus. et d habetur per F, Et cum de etiam habetur e et d per f et 
4
et
B
C4
F: C
=t C
seu fit
9F - t =
 ut C, etsi habeatur
tostat ut hima aequatio br + ECF + 2de = 2 1 (ab +CC, b (+ fat - tC, sc- ps, Ys: AtC
seu dd = _ 4, q tc, sc -pf, (F: C): Et, ubi pro f: c subiatri+ est quantitus inventa
= PF. fit e = +C, : pf) d = 4, 9f - tc, sc - pf,: pt d. Ergo dd =
porro ee = dd, sed et = -d sed Hinc sequitur etiam fre si D = - 1 fore etiam F= 
sed ita hunc valorem conjungendo, com aequatione in qua f: c assurgit ad quartum gradim
orietur nimua dete minatio, itaque non licet cere D simul et q facere aequale ihilo
Relior erit depressio, si fiat ab +ce =, b(+ f vel saltem si ambae sint in ratione
constanti e
ita enim cessabient, e exaltationes ob irrationplia, nis sublptis
+ at un xX L 2an9X + à 55
+ qu (bp + 2cd)
4, ab +cC
superesttertia aequatio
es
sis
9 terse B
pt
ct